What makes a physical constant truly fundamental? Does relating one
constant to another means that we can eliminate one of the constants?
First of all keep in mind that everything we, humans, describe is all based on our interpretation of the universe and we try to formulate it is such a way that we can best understand and describe how the universe works. In other words allow us to predict as exactly as possible what should happen given a set of initial conditions.
A fundamental constant is described as a value that is constant over time and space. A better term is universal constant. Basically if you set up your experiment correctly you should always measure the same value, now, in the past, in the future in any location of the universe. (as long as you take space-time warping into account in your experiment/calculations)
However not everyone finds this a satisfying or good description of what a fundamental constant should be. Some say that a fundamental constant should not be able to be expressed by other fundamental constants. In other words, every fundamental constant should only be possible to obtain by measuring it, even if you know all other constants. All other things should then be possible to calculate from fundamental constants and a complete theory of how the universe works.
The problem with that is, how do you decide which constant is fundamental or not. let's look at the case of $\epsilon_0\cdot\mu_0=1/c^2$, how do you determine which two would be fundamental and which one would not be?
Objectively there isn't much of a difference between the three, so the choice would be rather arbitraty. Therefor that description is unlikely to be adopted any time soon.
So what is a fundamental physical constant or in better words an universal constant?:
A constant that is the same in the entire universe that doesn't change over time.
If these constants would be different then the universe would look completely different and it's highly likely life would be impossible, our universe/constants are therefor sometimes also described as a goldi-lock universe, which is an often used argument for people who believe in multi-universe theories.
Also keep in mind that just because an "expert" in a field says something, doesn't mean it's true. Experts also often disagree, especially the more complex the subject is. He's also talking a lot of "empty air" and mostly talking extremely philosophical and proposing HIS view/theory.
Just take a look at all the comments below already and how many relative downvotes he has 108/630 (20-7-2020), which is 17% of the people downvoted the video. However only a very specific target group will even watch this video, indicating that many of the people that watch it (strongly) disagree with him.
It feels to me more that he is proposing if you know equation X, then you won't need to know constant Y. However that's a bit ambiguous because that also works the other way around then. Basically he's bashing many physicists and saying that they are focussing on the wrong things. So take what he says and how he says it with an open mindset but questioning.