Trying to understand the horizon problem I'm trying to understand the horizon problem better. For the record, I have read this post and this post, which have been tremendously helpful.
My understanding of the problem is that even if we postulate that the universe at t = 0 + dt was completely uniform, statistical fluctuations (deviations from homogeneity) would lead to less and less homogeneity at various parts in space, and because those parts are eventually causally disconnected, they would not be in contact to be brought back into thermal equilibrium.
My question is, is my understanding here accurate? What would you change about what I wrote to make it more accurate?

Okay, Andrew's response has been very helpful and enlightening.
I also found this chat. It contained a helpful comment by knzhou, and I thought it would be important to leave it here as well. It reads,

Well, if you assume the temperature was originally uniform, then it stays uniform. The horizon problem asks how the temperature got that way in the first place. You can just postulate it is, but then you're not really explaining anything.
However, the horizon problem could just turn out to not be a problem. Perhaps a sophisicated theory of quantum gravity automatically ensures the temperature is uniform without causal contact for some reason. For this reason, the strongest evidence for inflation isn't that it solves the horizon problem (which wasn't even the historical motivation for it) but rather that it predicts features in the CMB very nicely.

 A: Imagine a trajectory through spacetime that begins at some point $P$ in space immediately after the Big Bang, and travels at the speed of light. You can think of this trajectory as being a photon if you like, except that we will not allow this photon to interact with any other matter in the Universe. You can think of this trajectory as telling us maximum possible distance that any particle could have traveled from $P$. I will call this trajectory an "invisible photon" -- invisible here just is to emphasize that we are imagining that this is a hypothetical photon that does not interact with any matter in the Universe.
As time evolves, two things happen. First, the Universe expands. Second, this trajectory gets bigger as the "invisible photon" travels more and more distance. Intuitively, these effects tend to cancel each other -- even though the photon is traveling further as time passes, there is more space for the photon to travel through.
We can make this more precise by considering two initial points, $P$ and $Q$, Then, after a time $t$, we can ask: will the invisible photon that left from $P$ right after the big bang be able to reach the invisible photon that left from $Q$. If so, we say that $P$ and $Q$ are in causal contact at time $t$.
With that in mind, here is the horizon problem. We $P$ to be a point such that the invisible photon passes through the CMB and hits the North Pole of Earth today, and we take $Q$ to be a point such that the invisible photon passes through the CMB and hits the South Pole of the Earth today. Now of course, $P$ and $Q$ are in causal contact today. However, in "vanilla" models of the early Universe without inflation or some another mechanism, $P$ and $Q$ were not in causal contact at the time that the CMB formed.
The reason this is surprising is that we observe that the temperature of the CMB is uniform to an excellent precision. Why two patches of the Universe that are supposedly completely independent, have essentially the exact same properties?
Inflation solves this problem by modifying the expansion history of the Universe, such that $P$ and $Q$ are in causal contact at the time of the formation of the CMB. Since $P$ and $Q$ were able to come into equilibrium in the early Universe, it is not as shocking that later on when the CMB forms, we observe that they are at the same temperature.
Here's an analogy: If you and I both made a cup of tea in our houses, then we each drove to some meeting point between us and compared our tea, it would be shocking if the two cups of tea happened to be at exactly the same temperature. It would be incredible that you happened to heat up your tea to the same temperature I heated my tea at, and that the cooling was the same on the two paths we took. But now let's imagine that I made two cups of tea at my house, and you took one of the cups and took one route, and I took another route, and we met up at the same place. Now it is much easier to imagine that the tea will be at the same temperature -- they had a common origin, and went through similar environments to reach the final meeting place.

I am adding some extra discussion from the comments.
If two points are in causal contact at time $t$, then they are by definition in causal contact for all points after $t$. Reversing that, it is also true that if two points are not in causal contact at time $t$, then they were not in causal contact before $t$. So when we say that points were not in causal contact were not in causal contact before the CMB formed, they had never previously been in causal contact.
Now, you might say, surely when $a=0$, all points must be in contact. Well... no, but this is a subtle point. One way to explain why is that we don't really expect we can trust the FLRW metric all the way to $a=0$. When the Universe becomes small enough for the temperature to be so hot that it is above the Planck scale, we expect quantum gravity effects to be important. So, you can imagine there is a smallest, finite $a$ for which we trust the FLRW solution, and define $P$ and $Q$ at that time.
A more mathematical way of saying this is that the time $t=0$ is not strictly part of the manifold -- it is a singularity. Another way to say it, therefore, is (without inflation) that there are points on the CMB for which no past-traveling null geodesics emanating from those points ever intersect within the FLRW manifold.
