I'm afraid that this is all just silly. There are several related theories of why most genes on the Y chromosome degenerate and none of them predicts that men will become extinct. To see why not, we need to understand why Y chromosomes degenerate in the first place. All theories of Y degeneration… hinge on an unusual feature of the Y: it doesn't "recombine." … [snipped-out explanation of how chromosomes work.]So evolution is going to save masculinity! Yay evolution! Also, speaking of which, one thing I've found vaguely interesting is that most of the diseases and disorders that are increasingly afflicting humankind simply aren't the sort of thing that can ever be weeded out by evolution. Alzheimer's obviously sets in only long after a person has reproduced. Same with any number of other genetic diseases that hit the elderly. Dunno how this all ties in—evolution can save the male sex but not rising health care costs? Yeah, that's it.
This is important because recombination, it turns out, makes it easy for natural selection to get rid of bad mutations. Put conversely, natural selection is somewhat compromised when dealing with chromosomes that don't recombine. All our theories of population genetics thus predict that the Y will slowly but surely accumulate mutations that have slightly bad effects. But the key point is this: the process I have described will not spiral out of control, yielding sterile or absent men, for two reasons. The first is that the evolutionary forces that cause genes on the Y chromosome to degenerate turn out to be very weak; that's why it's taken hundreds of millions of years for our Y to fall into its current state of disrepair. In contrast, the evolutionary forces that maintain male fertility—and that even maintain a 50:50 ratio of males to females in populations—are very strong. The latter, strong forces, overcome the former, weak ones, and males neither become sterile nor disappear.
Second, not all genes on the Y chromosome are created equal. Some, indeed the great majority, originally resided on both the Y and X chromosomes. Natural selection will often tolerate loss of this kind of gene from the Y since there's a "backup" copy on the X that can still perform the gene's functions. Other genes, though, now exist only on the Y. Natural selection will most assuredly not tolerate the loss of this kind of gene since no backup copy sits on the X. The critical point is that most of the male fertility genes now residing on the human Y exist only on that chromosome and there's no way that selection will allow their loss.