Your staircase kite winders have detached and if not addressed the stairs will eventually fall apart.
You are right, the housebuilder will always blame the buyer's carpet fitters first. But if the kite winders were glued and screwed as they should have been, then a few small tacks would not have damaged them. If this means taking off the plasterboard underneath, then they will have to do that.
If the area of your floor in question is that large (especially if it is in the middle of the room) then the joist are either undersize or spaced too far apart. It won't be a squeaky nail, as the floor boards should have been glued and screwed down.