A (possibly) better solution to the issue might involve increasing the complexity level of tools. For instance, I have no less than 12 different screwdrivers in my tool box at work. Flatheads, fine flatheads, large flatheads, fine, large, and square center phillips, and torx. I also have a bit driver kit with about 40 different options to choose from for my electric tools.
Creating a makeshift screwdriver, say one capable of disassembling basic furniture with nothing but the simplest of screws without nuts or any serious torque applied to them, should probably not require much in the way of complex materials. Heck, I could probably go cobble one together out of junk in my garage in about 15 minutes. Would it be very good? Of course not.
Creating a screwdriver actually useful for crafting? That’d be a bit trickier.
So why not have a third level of screw driving quality. Simply increase crafting requirements by one level, leave construction at level one, and have the following:
Makeshift screwdriver: Screw driving quality one. Requires scrap metal (1) AND rag (2) OR short string (2) OR long string (1) AND Tool with Hammering quality 1
Screwdriver: Quality two. Keep recipe the same
Screwdriver set: Quality three
That should help prevent breaking intended balance and still allow for early game deconstruction.