How to remove this fastener

You took the wind out of my sails with that last paragraph. I was going to suggest using a nut/bolt.

As far as it causing problems when the tabletop is lowered, you can recess the bolt head. Yeah, you could make the cavity in which the bolt will stay on the underside of the tabletop, but maybe you'll drill too far or crack/scratch the tabletop if you chisel it out. It could end badly... which is why you should instead house the bolt head on the top-side of the cross member.

Brown = table top
Black = underlying cross member

The one on the left is what you were talking about. The one on the right is what you want to do.

Also, use a washer under the bolt head, etc.

Damaged screws are extracted by drilling into the screw with a drill bit, then using a special screw extractor bit that is tapered and has threads which turn opposite to those of the screw.

The extraction process should be done slowly and carefully because the extraction process is more fragile than the normal insertion of a screw.

I found this to be a good write-up with pictures.