No, it does not. There may be a very reasonable explanation for stasis but this is not it. There is always competition between individuals and the law is survival of the fittest, not survival of those who go elsewhere.
Let’s say you have two birds. One is black, one is white. Which one is fittest? We can’t say, there’s not enough information.
Let’s say you have two birds living in a dense forest. Which one is fittest, the black one or the white one? Now we have an answer. The black one. Better camouflage in this case equals more fit, assuming of course that the birds are equal in all other respects.
Let’s say you have two birds living in the arctic, which one is more fit? Now it’s the white one. Better camouflage in this case equals more fit, assuming of course that the birds are equal in all other respects.
Let’s say that you have two white birds that mate and have 3 offspring living in the arctic. One is completely white like the parents, one has red flecks on his plumage. One has one wing that is shorter than the other. Which one is the fittest? The answer is the one that is most like the parents.
Let’s say that the best adapted bird, namely the one that has not changed, has three more offspring. One is snow blind, one is like his parents, and one is a particularly deep sleeper. Which one is the fittest? The one that is most like his parents.
This is an extreme example, but it shows that nature can select the same traits in successive generations just as easily as it can favour change.
One other point, you use the quote by Ernest Mayer to prove your point that there are living fossils that haven’t changed in millions of years. Well… you are both wrong:
www.sdnhm.org/exhibits/crocs/tguide/tgcrocs.html
The point above is a teacher’s guide for teaching about crocodiles. It addresses the “Are crocodiles living fossils?”
I am sure that I will be able to show you how any of the living fossils you care to mention are not in fact unchanged from their ancestors. They are remarkably similar, but as I’ve pointed out above, nature can select for stability just as easily as it can select for change.