Say a man in a sexless relationship issues an ultimatum to his partner that he will end things if there is no sex, and as a result of that his agrees to sex. Why should that be considered non-consensual?

It's transactional in nature, "do this or I will do this". If his partner agrees to such a demand then she would be consenting to sex in exchange for not being dumped. She may not enjoy the sex, she may not be happy about having sex, but it's crucial not to conflate enjoyment and happiness with consent. A prostitute may not enjoy having sex with her clients. she may not be happy doing her job, but what she is doing is by no means non-consensual.

It's common for people to agree to do things they don't like doing in exchange for some other benefit.

Swap out the actions for something like "quit smoking or else this relationship is over" or “start paying part of the rent or this relationship is over” and few would consider it problematic.

Why is sex so special that it warrants it's own set of rules? Especially in this day and age with widespread birth control where there are essentially no consequences to it?