I play classical piano and sometimes upload to YouTube. You're right that there's no easy way to say "the person making the Content ID claim obviously doesn't own the copyright to the work in question". This happens almost every time I put a recording of myself playing Chopin, Bach, Mozart, etc and I get a notice from BMI or DG, etc.
You'd think that YouTube could at least categorize those who perform and record unambigiously public domain works and advise people using automatic match software to be _very_ sure there's an infringement.
Someday if I have time and patience, I may try to see what legal options I have against one of these deep-pocket companies who are making false (and libelous) claims about the recordings I have made. They're accusing me of theft, and it's provably false.
What's annoying is sometimes I'm on the road and don't get to responding to the request for several weeks.