The casings should be undercut for the tile. In other words, the trim should sit on top of the tile. In your picture, only the stops appear to be short, and that is easy to fix. Not sure why wholesale demolition is being suggested for this.

The jams are short as well. This picture shows better

I would find a babysitter for my child, and make sure I am there watching while they work. Something sounds a bit fishy. There should be no reason at all, to remove the tile, since it sits on top of the tile.

He's trying to gaslight you.

Your builder is either clueless or blowing smoke. You don't need to remove the tile. As difficult as will be to patch the jambs and casing, I'd opt to carefully remove the casings, cut the nails holding the jamb and reinstall the doors so the jambs sit flush. And then have a professional do the touch up painting-- not whoever painted it originally.

It looks like there was a different type of flooring before, not to mention the close-up shots look worst than it is...This can be fixed without removing anything as @User said you will never know there was a gap if it's done right.

Is there a reason why the mitered door stops at the bottom? If not...Replace them both, and have them go down to the tile.

I doubt that patches can made to look "seamless" with less effort than removing the casings and reinstalling the door. Yes, the stops will need to be replaced in either case. The fact that they are raw wood and the rest of the door is painted is telling.

Whoever did that work on the trim is incapable of fixing it based on the detail and sloppiness of the rest of the trim. The fix will likely be worse. The closeups are a hot mess of haphazd work.

I'd caulk that undercut and leave the stop alone as it will be less visible when painted.

This is the best you are going to get out of these folks.

Spend some quality time with caulk and learning how to use it and tape to create a very clean look.

If you want the to try to fix it - ask them to not send the same folks as did the install.

And yes - he's lying about all the rest (removing the tile, etc) because he wants you to decide it's not worth it. And chances are - if you push - he'll ensure that it's not worth it - because he knows it will be worse than before. Or - you will have given him an opportiuonity to fix and then you can either take him to small claims to pay someone to do it right.