We don't know whether the universe is infinite in space. The universe reachable to us is actually quite finite. The age of the universe is ~ 10^60 Planck time, and the size of the observable universe is also 10^60 Planck length. log2(10^60) ~ 200, so with all the universe time and space, you could only brute force a 800 bits string, or 400 DNA base pairs. That's not a lot at all. You can't just shrug complex systems as "random".
The actual number of bits the universe could've brute forced regarding life is probably less than 100 bits. That's not a lot at all. Think about bootstrapping a compiler in a hostile assembly with just 100 bits to boot.
Darwin's natural selection only gets you going once you have self replicating molecules and a way to inherit properties.
If it takes even 100 pairs of RNA to bootstrap life, that's already too much.
Another funny way to look at it, is that if there's truly no randomness in the universe (if everything was predetermined in the big bang), literally everything you know has kolmogorov complexity of less than 1000 bits+number of bits required to represent laws of physics + number of bits to describe the initial condition in the big bang. The program that decompress it just simulates the whole universe and fetches the information from the given coordinates.