Standard evolution theory involves the concept of genetic death due to substitution, i.e. to old stock dying out BECAUSE of the appearance of a new and superior genotype. All Remine is doing is postulating the maximal possible value of such a phenomenon and demonstrating that even that cannot make a reasonable scenario out of the idea of man evolving from an "ape-like ancestor" in ten million years or less.
Remine's postualation is inherently flawed. Evolutionary changes aren't limited to one lineage. It assumes that all evolutionary change is linear and precludes any exponential changes.
I don't think evolution is entirely a matter of new genes appearing by mutation, then being selected. What if a new gene causes expression of several genes which are already present, but latent in the ancestor? Couldn't that produce more rapid evolution? There are probably many genes shared between modern man and chimps or ancestral hominids, which are expressed very differently in the various species. This could help explain why man and apes have so many genes in common, yet are phenotypically very different. Human evolution, for example, is postulated to involve neoteny, by which genes active only during early youth in the ancestors remain active much longer in man.
If this is your strawman version of evolution, it is no wonder you can argue plausibly against it.
However, biologists are not as stupid as you appear to be. New species do not appear in one generation. Evolution is a change in the frequency of alleles in a population over time. There is never a point in which one generation is a different species from the previous species.