It really makes no difference whether mutations are REALLY random or just seemingly random. They supply the stock of variations, and the range of observed variations covers the gamut of possible alleles. This is an observation based on numerous experiments.
It makes no difference whether some invisible entity is shading the dice. The results are indistinguishable from random.
What isn't random is the effect that alleles have on individuals and their reproductive success.
Yes, exactly correct. Given a large enough population size and enough generations EVERY possible single nucleotide polymorphism (single DNA base change) will be generated by mutation and selected for or against by the environment. In that context the term “random” loses its meaning.
It is like trying to pick a combination lock by going through every possible combination. The success of this endeavor is guaranteed, it just takes a bit of trial and error and time. The eventual success of this tactic is not in any way “random”.