Does a bad screw require changing the tap itself?
The reason why the plumber says this is that the cost to replace just the screw (for him) would be greater than the cost to replace the whole handle and valve assembly. This is because he can replace the whole handle assembly in 10 minutes, but it would probably take him at least 30-45 minutes minimum to find and install a perfect replacement screw.

If you value your time less, feel free to look for a replacement screw. Note that the screw will have to be a certain length and thread pitch and diameter etc.

If the internals of the valve are cleaned up, resurfaced and washers replaced, the valve should operate correctly. The screw is for the handle and is not part of the valve. 

Getting that screw out, by the way, will be annoying. The way to do it (not involving high temperature processes) is to use a sleeve bushing around the screw, a drill bushing inside the sleeve, then drill a hole for a screw extractor.