How would you fill the gap at the top? I'd say no.

It would be much harder to go over it and make it look right than it would be to remove it. Much of the surface damage to the drywall could be repaired just enough to make it flat, not flawless and covered with the new baseboards.

I'd say there is no way to go over it and have it look good as you'll be trying to cover the gap and that is going to look patched.

Carefully pry the tile off and use the right tool - there are pry bars made for that type of work that if used carefully will leave very minimal damage.

Then buy taller baseboards than what are there now and that will cover the potential damage to the walls of removing the original tile.

Once the baseboards are up and caulked and painted you will be much happier with the overall result than if you try to go over the top.

Don't cheap out, do it the correct way.

No is the answer and you have good advice as to how to do it the right way.