Should I add a plywood subfloor below my damaged hardwood floors when I reinstall?
Two "minor" points... if the plywood is going to genuinely offer support, it probably needs to be at least 1/2".  Have you thought about whether that's going to create other problems with clearance, such as at transitions/intersections/door bottoms?  Also, assuming that the flooring runs perpendicular to the existing framing (joists etc), if we're really talking about small slivers, such as you describe I'm not at all sure that you need to do anything.  Is there no existing subfloor, the flooring is just attached to the joists?

Can you post pics of the typical damage?