Did horse and cart operators ever get their jobs back?

I mean... yes.

Blacksmiths, wheelwrights, and wainwrights transitioned to auto mechanic or carpentry work. Not all of them, but for most the skills were portable. Several of the carriage-making businesses transitioned to automobiles. Studebaker and Oakland (which became Pontiac), for example. I'm sure the most experienced people lost big, of course, and we shouldn't minimize the damage done to people's livlihoods.

Unless you mean taxicab drivers, which was the term for them before the automobile. Taxicab = taximeter cabriolet, and a cabriolet is a style of horse-drawn carriage.

It really was the horse people that lost out the most. That's why the idiomatic reference is for buggy whips, not so much for buggies themselves.

This is unironically the best comparison.

maybe they became taxi drivers. But that too will be gone soon.

Can't beat capitalism + technological innovation.