Replacement caliper assemblies aren't stupid expensive. As long as you have it apart, replacing the assembly is the easiest way to go.
If the caliper was overheated then the seals in the cylinder may have been damaged. Replacement is a good idea. If this was regular maintenance at say 150,000 miles with not brake issues I might say replace the seals yourself. But if it's already damaged? Probably not.
Rotor might be cooked too. If so, rotor and pads need to be replaced, which means both sides.
If it's really just a sticky slider then you can disassemble that and grease them up.