Why am I having bluetooth problems? Or is it something else?

Well this is weird bc Bluetooth is supposed to be backwards compatible.

The SE1 supports Bluetooth 4.2 & newer Garmin watches supports 5.0 but it should connect to your phone, although it would be running at 4.2 standards. I'm guessing if music plays, airdrop works, and everything else functions it's probably related to software. Does your Garmin app tell you an older version of the app will be downloaded when you download it from the App Store? Some apps doesn't support older iOSs and would automatically downgrade themselves, so the older version might not be able to work with the new watch. Another possible explanation is that the app itself requires 5.0 to function so it won't work

Same issue with the school gate app, although you might want to make sure that doesn't rely on NFC to function, as the SE1 does support it but can only use it for Apple Pay.

Some things you can try is to look at Software Updates in settings to see if there's an update available and reinstall both apps. If that doesn't work then unfortunately the SE1 might simply be too old for modern tech.