This reads to me like wanting to learn sports, and then disliking that your soccer instructor didn't teach you how to throw a ball. There's a lot you can learn about music in the classical framework, but the focus there is certainly not improvisation based on chord sheets. Generally learning about improvisation in a high school context would happen in jazz band, although in my experience lots of people in jazz band (including myself) are not too interested in improvisation either. A separate music theory class might also help although dedicated classes for that are probably even less common than having a jazz band, although if a student is particularly motivated and interested and the school is at all flexible, you can sometimes work out a deal with the band teacher to do a special run of such a class. Also music theory is just theory, it's totally insufficient without also actually practicing improv as doing it in real-time is a much harder challenge.
I wouldn't doubt if learning guitar or piano would help with this aspect, but I know plenty of people who learned it just fine on wind instrument if they want to. It's not too surprising especially for those learning wind instruments - taking a solo in high school is already scary for a lot of people, and even more so when they don't even have music to play or practice ahead of time. Also there is admittedly probably a higher barrier of entry to improv on wind instruments so a lot of the focus necessarily has to be on achieving a baseline technical proficiency before it even makes sense to think about improv (if they don't put in the prerequisite technical work, it would be hard to move to improv). For example, with guitar or piano, right off the bat I can play any note in tune and sound at least halfway okay. On a wind instrument, even playing the most basic note will sound quite bad at first, and generally the range of frequency is limited for beginners and learning to extend it can take many years. Playing in tune likewise takes years.
Anyway, all that said, I entirely agree, if someone really likes music, it's hard to go wrong with also learning guitar or piano. Piano in particular, as one of the most versatile instruments, used in so many genres, and can help with learning fundamental theory, composition, and even gives a very good sense of rhythm as the two hands have to act independent of each other in a way that other instruments normally don't have to. If I had to go back and learn a different instrument, piano would be a likely candidate.