we pick the MRCA semi-arbitrarily based on criteria like "sexual compatibility of descendents".

"I know we're both humpback whales, but he's nowhere near as adventurous as I'd like him to be..."

I think species can be paraphyletic. If we sent a family of llamas into outer space and they evolved into Space Llamas, there would be no common ancestor which included all terrestrial L. glamas but excluded L. astrollama.

And there are groups (like oribatid mites) where parthenogenesis is very common. No sex at all, though males occur. (Here's a challenge: you think of anything common among vertebrates, then look for invertebrates (including single-cellular animals) for whom it's not common. The sea-dwellers are very good for this search.)

Some people would tell you that only Homo sapiens exists as a species. Suppose a 'species' exists as a set of disjointed populations, which will never meet each other (or the probability of it happening is so much smaller than of them going extinct)...

No, that's a clade or a monophyletic taxon. Most species are clades, but as solipsist points out not all species are necessarily clades, and most clades are not species.

There are various genetic issues that make individuals sterile. We don't say that they are suddenly another species just because they are sterile and thus not sexually compatilbe.