90% of damaged screws are fixable by peening the metal back into place.
Those that aren't, I lathe turn the damaged top off down below the slot. Silver solder a new oversize piece of round stock in place. Turn that to the orig screw head size.
I leave the head 'tall' and with a rough hacksaw slot.
Turn the screw into place a few times to seat it then mark the 12 & 6 slot and surface lines.

Take the tall head down to just shy of the surface mark. Use the 12& 6 marks to cut a new narrow slot with a Jewelers Saw.
I usually back the slot up to 11 & 5 as that one last time back in place will always draw up tighter than all the ones before it.

Final trim the head to the surface or round as needed. Polish and engrave

The Hard solder joint is well below the surface and not seen.
I've done hundreds like this. It goes pretty fast once you're used to it.
Re-capping screws...

Welding always seemed like a PITA way to do it.
Bad welds, hard spots, etc.