Yes. The district should supply the Chromebook for school work. They will manage that as they see fit. If he wants to do other stuff with a Chromebook, he should have a separate Chromebook and separate Google account. Ultimately that's easier and safer than constantly logging in and out of two different accounts on one machine anyway.
I am not a parent, but this seems like a good practice to get a child in the habit of, anyway: separating out your devices for work and school. Much like I wouldn't log into personal Slack groups on my work laptop (I learned that lesson!), I wouldn't try to conduct personal work on a school laptop.
> The district should supply the Chromebook for school work. They will manage that as they see fit.
I haven't been a student for a while, but the closest analogous technology they had when I was in high school was my graphing calculator for math classes. The school district mandated individual students each obtain a specific graphing calculator, which was a fully programmable computing environment (the TI-83). But the teachers/administration could and would wield a lot of power over those devices (which the the families owned in the legal sense) - looking through it, requiring students to wipe its storage with no warning, etc. Requiring families to buy a Chromebook and still use a school-managed account with invasive management on it feels largely analogous.
Buying a separate personal laptop is the correct workaround, but unfortunately I don't think the "the school should supply the schoolwork computer" line of reasoning holds up. The hardware/software has become more powerful in the graduation from TI-83 to Chromebook but the principles are the same.
In my case, we bought our kids better ones than the district offered, which are the lowest educational spec machines available. It was only after we bought it - during covid school-from-home last year - that we learned our district also forbids any non-district-issued computer from connecting to school wifi, so we ended up with one of the crappy machines anyway. On the plus side, no effect on our personal chromebook, but on the negative side, my kids are restricted to using the crappy school computers for school work.
Chicago Public Schools gave me a chromebook to do homework on. It was a piece of trash computer that they probably got for free, but I could certainly take it home and use google suite for homework.