I'm unwilling to assume it "just randomly looks like that" given that a random distribution on a universal scale seems highly unlikely to show the large scale structures notable in the CMB.
So for instance the CMB did not support the big bang, it contradicted it. See: the horizon problem [1]. To reconcile this inflation theory was invented which arbitrarily suggests that the universe hit the accelerator hard, then slowed back down. There's no logic, mechanism, rationale or falsifiability. But it makes what we see fit what we predicted we'd see, so it's a pretty widely accepted part of modern physics. And this retrofitting now has a cascade affect that enables other theories to provide support for yet other theories -- such as what you're proposing here in that the CMB now 'supports' the big bang. It does so only if you add a very big asterisk there.
If you'll reread it you'll see that I mentioned nothing about a Big Bang, but rather that the CMB provides evidence that the Universe expanded from a smaller state to its current state, which is larger than its starting state.
From the linked wiki: "Differences in the temperature of the cosmic background are smoothed by cosmic inflation, but they still exist. The theory [Cosmic Inflation evidenced by the Horizon Problem] predicts a spectrum for the anisotropies in the microwave background which is mostly consistent with observations from WMAP and COBE."
But physically what we observe does not make any sense. Like you probably know, nothing -- including action -- can be perceived to travel faster than the speed of light. The sun is about 8 light minutes away from us. If it somehow just suddenly disappeared, we'd still see it in the sky and continue to revolve around, what would 'now' be nothing, for about 8 minutes. The observation of its disappearance and the effective causality of its disappearance (and its effect on our orbit) would happen at or very near the exact same time.
The problem with the CMB is that areas of space that should not be causally connected since light itself has not had time to go from one to the other, seem to be causally connected. In other words, with our boiling pot in a kitchen room - the eventual equilibrium that the kitchen reaches (if we assume that that entire little region is all of the space in existence) is going to vary quite substantially whether you have an e.g. 100 cubic meter kitchen or a 200 cubic meter kitchen. We should observe both sides of space acting like two independent 100 cubic meter kitchens, instead we seem them behaving like a single 200 cubic meter kitchen.
This is a major and unresolved problem that threw much of what we know out the window. It directly contradicts the big bang. To 'resolve' this, we started creating a arbitrary special conditions. Cosmic inflation is one of these. There is absolutely no reason to believe that cosmic inflation ever happened - its sole and only reason for existence is to work as a 'fix' to make what we observe fit what we thought we'd observe. This makes it illogical to use derivative things as "evidence". In particular the nature of our current CMB is in no way meaningful evidence of inflation, because inflation was hypothesized, after the fact, in no small part to fit the CMB to what we thought we'd see! In other words calling the CMB meaningful evidence of inflation is trying to provide support to a hypothesis by suggesting that the observation said hypothesis tries to explain is meaningful evidence of that hypothesis.
Any not completely idiotic hypothesis will obviously be 'evidenced' by what it tries to explain. But we have a major problem when that 'evidence' becomes all you have to rely on, and that is exactly the case here.