Nothing is stuck in the frame, but the bolt sections going in are very tight.
These calipers have to be turned to get them to go in. They won't push in.
it [a recessed nut] could be seized in the frame as I had a few years back
I suggest that the very fact that "the calipers have to be turned" and they "won't push in" means there
is 'something' "stuck in the seatstay 'bridge' hole.
We can see the threaded nut inside the 'ole, no? See
@biggs682 experience.
I don't think the OP will be able to extract this errant nut by 'unscrewing it' with an allen key per
@fossyant : why would turning it have any extraction effect? Might be a useful 'loosen' though.
Get a bolt the same OD (?M5) and tpi as the one on the caliper, screw it in 'as far as' and 'tap' with a rubber mallet, supporting the seatstay securely.