Why would a person train exclusively in boxing if they wanted to fight MMA (less rules then boxing)? Why would someone train MMA if their goal was self-defense (no rules)?

A boxer may do well in MMA, and an MMA guy may do well in self-defense. But if your goal is only self-defense why train for a sport? (unless you just like the sport)

Because those sports consistently develop functional skill for exactly the same thing that TMA is supposed to do - fighting. And does it quicker. There is virtually no functional difference between defending a punch from a MMA opponent or a punch from anyone else. The skill is the same.

Your argument rests on the dubious possibilty that non-practicable TMA skills would reliably be able to be implemented under pressure. Something that the early minimal-rules UFC's have shown to be unlikely.
"In case you ever wondered what it's like to be knocked out, it's like waking up from a nightmare only to discover it wasn't a dream." -Forrest Griffin