So with your "tool: lute" proficiency when the DM says you need to judge between two lutes to find the better one. What do you do? My point here is, there are probably many uses of a tool where not actually having the tool in hand could be a thing a DM could make a call for. Treating them as tools you either have to overlap with skills or you simply have those potential rolls vanish or perhaps get rolled into just a stat check. Breaking them up seems to cause more rules to appear when less is usually better.
Intellegence (Appraisal) Check!
Also, if you are judging 2 Lutes, do you not have 2 tools with which to make your judgement?