I told you, I only get 10 minutes at a time. I run a business, I'm not in a nursing home. The best time to catch me is late at night when I'm too tired to sleep.
I wish I had time to educate y'all about math. Some of y'all don't even seem to know what "random" means. Two of the most important words in science are SCALE and SCOPE.
"WHY" do proteins evolve? I don't know, they just do. I can give you mechanisms but that might not answer your question. Proteins probably evolve for the same reason DNA does - combinatorial complexity. Although the "success" of a protein is defined a little differently.
See if this makes sense to you - a biological (molecular) "need" is an attractor. Selection works by following trajectories along attractors. When conditions change (either mutations or niches) the attractors change too. An attractor can be stable or unstable. For instance there are about 40 or so proteins that "never" change, they're practically identical in all life forms. And there are some that "deliberately" change, they're programmed to change by the DNA. These latter ones "evolve", rapidly. If I have time I'll find the Dutch study for you, that looks at protein evolution in roadside weeds. It talks a lot about post translational modification, which is an evolutionary path that doesn't necessarily depend on DNA.
How rapid evolution is selected is still a mystery. In our own brains there are some areas that are evolving rapidly and others that are highly conserved. It "seems like" the evolution is occurring for a reason, and has a goal. But we probably won't find out in my lifetime, it takes longer than that just to observe the process.
I can only address "why" mechanically, not metaphysically. Ultimately everything boils down to universal symmetries, "why" those are we can only guess. Everything we see around us is a direct result of those symmetries, including the shapes of proteins. If I have time I'll talk with you about randomness, and what exactly it means. Protein evolution is distinctly non random. There's very little randomness to it. It is carefully controlled in the cell.