I used heat, and some proper bridge removal knives, but there was still some tearout on the guitar top. It's not terrible, but not good either.

I'm going to make a new bridge with just a bit larger footprint than the old, because in my opinion the old one had too little material ahead (nut side) of the saddle. I believe this is why it failed, cracking on both ends of the saddle slot.
I will remove the finish and clean up the whole area under the new bridge so the glue is on wood.
To get back to my problem: Do I have to fill the area where the wood is missing with something?
Or, will the hide glue itself fill the gap?
I will take a picture and post it later today.