Checks to do with music instruments are all Performance checks. Leading to the total roll being:
D20+CHA Bonus+Proficiency Bonus (since the bard is proficient in the instrument)
To address your last point. All checks to deal with musical based instruments are Performance. Yes they are tools, but you have to know how to play them, and know how to play them well. That's where the concept of being proficient with them comes into play.
A thief who is very good with his hands could have the finesse required to play the piano for example, but canwas never trained in playing the piano (Not proficient in piano). So chances will be he won't play it well? No.
Not all tools utiliseutilize the same skill in order to accomplish the task they are intended to do.