The random mutagenesis model strikes me as rolling dice. Now, maybe time is sufficient to get enough rolls to come up with new species... but look at all the species that have either existed or currently exist... that's rolling a lot of sevens.
Where I think evolution overreaches in in the presumption that we have a good handle on the mechanism for it. Maybe it's random, but I don't see how that's confirmed in the time frames we've been observing living species scientifically.
I think we use the word random too often. At the fundamental level (where it really matters) the chemistry is not random. Chemical bonds and long chain polymers follow a very distinct set of "rules". But remember, biochem is not my formal educational background. So if I have posted in error, I hope I will be corrected.
Problem with mutations is that they cannot occur fast enough to achieve what the evolutionists say they achieve. For example, lacking all other proof, the evos have resorted to calling the building of resistance to chemicals, and medicines by viruses and pests as evolution when clearly it does not prove any such thing. Let's look at the situation in a logical manner. The dead do not reproduce, so clearly the viruses and pests killed by these medicines and chemicals are not the source of the genes which have become resistant to these medicines and chemicals. The source of the genes for the resistant strains of these creatures must therefore be found in the creatures that were not killed by these chemicals and medicines. Since these creatures managed to survive the medicines and chemicals there was no mutation required for their successors to be resistant to these medicines and chemicals. All they needed was normal reproduction!
The players have been rolling dice for lo these billions and billions of years now. There are trillions and trillions (maybe even quadrillions) of players too. The adequate survive.