Why is the quantum Venn diagram paradox considered a paradox? I've just watched this video on YouTube called Bell's Theorem: The Quantum Venn Diagram Paradox

I don't quite understand why it is considered a paradox 
At 0:30, he says that as you rotate 2 polarizing filters, less and less photons come through the 2nd filter (and when the angle is 90 degrees 0% of photons come through). 
Then at 0:55 he adds the 3rd filter in the middle rotated by 45 degrees and expect two 45 degrees polarizing filters to be equal to one 90 degrees filter.
But 45 degrees filter + 45 degrees filter (relative to each other) does not equal to one 90 degrees filter, right? why does he expect it to be?
I don't understand why he's so surprised saying "somehow introducing another filter actually let's more light through" at 1:05. What's so surprising about it? 
He rotated the 2nd filter by 45 degrees, allowing photons with polarization of 0 - 50 degrees (since 45 degrees is 50% of 0-90 degrees range) to come through. Why would they have any problems coming through the 3rd one which rotated 45 degrees relative to the 2nd one?
At 1:10 he says, "the more filters you add the more light comes through". Well, no kidding, by adding 89 more filters between the first filter (0 degrees) and the last one (90 degrees) you just widening the polarization range photons can have - you're basically allowing 100% of photons from the 2nd filter to go through to the end. So in theory, using perfect filters, the amount of photons you can see will gradually increase, and by the time you add 89 more filters (with step of 1 degree) you'll be able to see the same amount of photons as you could after the 2nd filter.
 A: Not a paradox but it is considered surprising to many as a more natural Venn diagram is expected as explained there, a better explanation is on this part of the follow up video, which as my understanding, adding an intermediary filter causes an actual change to the wave which helps more amount of light to pass from the latter filter, which if you are familiar with measuring effect of quantum states, it can make more sense.
I've picked two screenshots from the given link which I guess can help for a better understanding, first, portion of light which passed the first filter:

After that, the change that happened to the passed light after hitting the second filter,

Which this change helps pass of more light from the latter filter, very similar to what happens if we do measuring on a quantum state.
Of course this explanation, without considering the actual debated meaning of quantum state and measuring, can be considered very erroneous from various aspects and have assumption that still is under controversy or nearly disproved (hidden variables) and even different quantum interpretations can give different reasonings to the very same phenomenon as far as I know, I guess however it is somehow can be acceptable for a limited scope.
A: 
He rotated the 2nd filter by 45 degrees, allowing photons with polarization of 0 - 50 degrees to come through.

This is where your misunderstanding lies. 
The first filter takes away all the photons' y-component. The last component takes away all the photons' x-component. The paradox comes into play when you treat the incoming light as particles (photons).
A: To clear your mind I want to tell you in detail how the filters influence the light.
A polarizing filter (for some range of light) let 50% of the incoming light through the filter. Behind the filter the light is polarized: the electric field component of all photons is oriented in the same direction.
It is important to understand that this 50% polarization happens even for light from a thermal source. The light from a thermal source is unpolarized, means the direction of the electric field component of the emitted photons is equally distributed over 360°.
[
Would you agree that the filter rotates 50% of the light from the thermal source? In the case of the filter from the sketch photons oriented between +/- 45° and between 135° to 225° get rotated and are polarized behind the first filter.
To proof this one takes the same kind of filter behind the first but with 90° rotated to the first: no light comes through. So one get the proof that this filter really could not rotate photons with field components which are perpendicular orientated to the filter. Rotating the filter more and more light is going through. Dependent from the narrow band of the filter the relationship between the angle of rotation of the filter and the light intensity varies.
What I've ever seen were so narrow band filters that light was going through the second filter only if this filter was in the same orientation as the first filter. So the sketch is a idealization, in reality the light has some variation angle of its polarization.
Now you can understand why a third filter between the first and the last with an orientation not equal the two others let light through. Simple the second filter rotates the light in the same way as the first filter do. And the last do so.

Why would they have any problems coming through the 3rd one which rotated 45 degrees relative to the 2nd one?

So your intuition is right, there should be no doubt.
A: The surprising part only shows up when you have entangled photons and make measurements far apart
Up to the half of that video, they talk about a 1-photon through 3-polarizers experiment.
But as clarified half way through the video at https://youtu.be/zcqZHYo7ONs?t=526 , the single photon case is not really surprising:

Drawing those Venn diagrams assumes that the answer to each question is static and unchanging. But what if the act of passing through one filter changes how the photon will later interact with other filters. Then you could easily explain the results of the experient.

The rest of the video then continues to explain the actually mind-blowing part of it: if you do an experiment with:

*

*a pair of entangled photons

*two detectors separated far apart, one measuring each photon, and separated further than what the speed of light could travel between measurement times/locations

then one measurement still seems to affect the other. And this time, there is no way in which the intermediate measurement could have affected the state of the other photon.
And then they clarify that everything before that point was just an introduction to polarizers, which is especially useful because the maths are the same for both.
While the Venn diagram helps to organize the idea of "suppose each photon knows what it will do for each filter", I think that it is a bit hard to see the actual counting argument from it. Maybe we just need better Venn diagram drawing skills.
Recapitulating the experimental results of the entangled photon experiment
Angles used:

*

*A: 0°

*B: 22.5°

*C: 45°

Percentage of time that the result is the same for both photons (both pass, or both don't pass):
Angle 1 / Angle 2 / Probability

AA cos(0°) ^2 = 100%
BB cos(0°) ^2 = 100%
CC cos(0°) ^2 = 100%

AB cos(22.5°) ^2 ~= 85%
BA cos(22.5°) ^2 ~= 85%

BC cos(22.5°) ^2 ~= 85%
CB cos(22.5°) ^2 ~= 85%

AC cos(45°) ^2  = 50%
CC cos(45°) ^2  = 50%

Or in a more condensed form:
angle difference / results same
 0     100%
22.5    85%
45      50%

which shows the intuitive idea that the closer the angle, the more likely is the result to be the same.
Notably, at identical angles, we have the very strong guarantee that the results are always the same.
Therefore, we don't need to say all the time:

Photon 1 would pass A and B but not C, and Photon 2 would also pass A and B but not C

because if photon 1 would pass A and B but not C, then necessarily photon 2 would also pass A and B but not C, because we have never seen a single experiment where we get different results at the same angles.
Thinking about the probability contradiction
Now, suppose that at photon creation time, each photon has some inner state that determines if they would pass in each angle (realism).
Or in other words: the randomness of the results does not come from a fundamental randomness of nature, but just from random variations of our experiment which we cannot control or observe due to technological limitations, e.g. something like the exact position at which an incoming photon hits an electron of the crystal before splitting into a pair, or the exact state of that electron at a given point in time.
If this is true, it makes sense to ask the intuitive question:

Would a given photon pass a given detector?

for all given angles, even if the angle is not set during a particular experiment.
We will now see that this is impossible given our experimental results.
Let's introduce the following notation, where upper case means pass, and lower case means no pass, e.g.:

*

*AbC: photon 1 and 2 would pass A, would not pass B, and would pass C

*aBc: photon 1 and 2 would not pass a, would pass B, and would not pass C

And let's denote an unknown state by not adding it at all, e.g. Ab means:

pass A, does not pass b, and not sure about C

Since we only have two entangled photons, our experiments can only tell us about statics involving pairwise results (AB, AC and BC).
So the trick is to make an inequality using the pairwise information that we have:
Ac <= Ab + Bc

which is impossible because experiments tell us that:

*

*Ac = 50 since separated by 45 degrees

*Ab = 15, since if A passes, usually so does the nearby 22.5 degree B

*Bc = 15, since if B passes, usually so does the nearby 22.5 degree C

and thus:
50 <= 15 + 15

The inequality can be seen directly the Venn Diagram, since the region Ac is entirely contained in Ab union Bc.
But we can also obtain it algebraically by expanding Ac, Ab and Bc into their constituents (the smallest regions of the Venn diagram):
Ac = ABc + Abc
Ab = AbC + Abc
Bc = ABc + aBc

and then:
ABc + Abc <= (AbC + Abc) + (ABc + aBc)
0   + 0   <= (AbC + 0  ) + (0   + aBc)
0 <= AbC + aBc

since AbC + aBc must be positive.
So our premise that photons know in advance what they will do must be false.
It does seem that the decision to pass or not a given polarizer is a fundamental randomness present in nature, not a limitation or our experimental technology!
But then how is it possible that they always give the same results on identical angles?
The only solution seems to be something along: when one of them reaches the measurement device, it somehow communicates the other what it decided to do faster than light (non-locality), so that the other will do the same.
Or something more complicated like "both photons communicate with each other and with the polarizers at all times" (and therefore faster than light).
How are entangled photons generated after all?
I think it is always useful to have the basics of the experiment in mind.
In the case of photons, there is one extremely well known and relatively easy/precise way of doing it: spontaneous parametric down-conversion.
You basically just shine a laser at a special crystal, and at some very rare events, an incoming photon gets converted to an entangled pair coming out at a weird angle, unlike the rest o the beam that goes straight. And you calibrate the laser intensity so that you don't get more than one entangled pair at a time, so you know for sure which photon is entangled with each photon.
This is a great video about it, which actually shows a optical table generating such photons: https://www.youtube.com/watch?v=1MaOqvnkBxk
Since photon pair creation time is random, in the experiment we are only able to observe events where at least one of the photons passed through one of the filters. So what we do is to count:

*

*did we get two hits at a given time (coincidence)

*or just one (non-coincidence)

and double blocks (aa, bb, cc) are never truly directly observed.
A: I think the most intuitive answer for why this is a "paradox" is because it behaves differently than other types of filters such as color filters. In the case of non-polarized filters, the behaviour is entirely subtractive. The more filters you add, the darker the color and the less the amount of light let through.
For instance, consider the following set of subtractive filters:

Compared to a similar filter, as in the image in the question's link, you can see how it's unintuitive that the center of the "Venn diagram" is lighter than some of the overlapping cells.

Obviously, the video itself tries to explain this paradox in more detail. For myself, it seems very counter intuitive that something that reduces the amount of light can actually increase the amount of light when put between two filters.
