I do disagree slightly with the cajoling into playing instruments. Now, my experience is just that- mine, so I'm not saying you're even probably wrong or right. My brother, sister, and I were all cajoled into at least trying music when we were young.
Our elementary school did mandatory xylophones in 2rd grade and recorders in 3rd for all the kids. From 4th grade on you could opt in to playing any instrument, though if you needed to play a school instrument is was the usual suspects of violin, flute, clarinet, sax, and trumpet. If you already had access or were playing one you could bring your own.
I was gonna bag it and not opt in. My parents (strongly) suggested I pick one for the year and give it a whirl. I tried the violin and sucked so very badly. Mid year I switched to clarinet because that's what was available and I was going to quit entirely if I had to continue with violin.
Side note- my sister played as well and has gone into software development after excelling at school and university, specifically science. My brother did terribly in school, didn't graduate, and gave up on music on about day two (he's wound up being very successful any way though).
By my 8th grade year I was an extremely accomplished clarinet and sax player (first chair, solos, competition band, marching band, performed on some recorded albums, etc.) and I swear I started feeling dumber after I gave it up. I played in my very limited free time in highschool as I was way more dedicated to sports and our HS band was good enough that you could do sports or band, you could absolutely not do both.
I looked into it and there are a few studies that suggest people that participate in playing instruments often do better in school and are more mathematically inclined. I'm lacking sources as I'm on my phone.
I agree that most busywork homework is bullshit. I'm okay with the monthly/bi-quarterly assignments that require free form thinking and finding your own way on a subject like research papers, craft based projects, book reports and such. Anything that lets my child delve into their thoughts I'm okay with, as long as they get a sufficient amount of time that they are able to go deep, and not just churning out crap.
I will absolutely cajole my kids to try a plethora of instruments. If we get through strings, woodwinds, percussion, and brass and they haven't found a winner- then we can bag it. I expect it'll be two or three years of trying. If they seem like they really hate it or we've exhausted our options, we'll let it be. Given the amount of music we surround them with now, and how they interact with it (drumming, Casio keyboard, dancing), I think we'll find a way to keep them involved in music.