My dd (10) did a music taster day at school, and jas come home wanting to play the flute. I refused to let her brother learn an instrument, because I think the lesson of being considerate to others and realising that there will always be people with more opportunities than them is much more valuable than any questionable benefits of learning an instrument.
Also, she tried gymnastics at age 8 and she gave up after a year, so we will probably just piss off the neighbours for nothing. She does swimming as an extra curricular activity already and that night is a huge stress for everyone with dinner being late. Surely doing more than one will interfere with her studies due to her being out routine.