Let's say Bob the Bard is proficient with the lute and the Performance skill. He wants to sway a crowd by playing a melody with his lute. What is the correct roll in this case?
Option A. That falls within the purview of the Performance skill, so it should be a Charisma (Performance) check.
Option B. He's trying to emotionally affect a crowd, so it's definitely Charisma, but he's using a tool he's proficient with to do so, so it should be a Charisma (lute) check.
Option C: Both options A and B are functionally identical—just roll a d20 then add Bob's CHA modifier and his proficiency bonus—so the question of choosing between A or B is moot.
Option D: Playing a lute is clearly an act of precision, finesse and hand coordination, so it should be a Dexterity (lute) check. This matches the real world expectation that you don't actually need to be a sociable or charismatic person for a crowd to like your performance, you just need to play your instrument well.
If you chose Option A: It makes sense for the Performance skill to cover acts such as poetry, comedy, acting, storytelling, etc, that don't require tools, but if Performance covers even musical instruments that are tools, when would you ever make a musical instrument check, in the way a rogue makes a thieves' tools check?