So I bought a Dremel tool and figure I'd cut off all the tile that way,
then replace the drywall. Can anything think of anything particularly
stupid about removing tile this way?

You can use the dremel to cut the wall board above the tile and then pry the
whole mess off the studs. If you think the dremel is going to cut the tile
or the grout you are about to be disappointed.

The low tech way is to start at the top wearing heavy leather gloves sing a
flat ply bar and a hammer.

Colbyt