revert from Monterey to Big Sur OS

If you don't have a Time Machine Big Sur backup, reverting back to Big Sur isn't straightforward. You can wipe and install a new Big Sur system, but you will have to manually drag drop over files from your Monterey Time Machine backup to get things back. If you use Apple Mail, Photos, other Apple apps, the formats may have changed from Big Sur to Monterey, adding more issues.

One thing to try might be:

(1) Make a second verified backup, separate media

(2) Follow Apple's instructions to restore to factory settings. These vary according to your model. Install Monterey.

(3) On first boot up, create a new administrator user, say ADMIN, but do not migrate anything over. Check bluetooth. If it doesn't work you have a hardware problem.

(4) Update Monterey to the latest version. You still have just the ADMIN user, nothing else but a factory fresh Mac with Monterey.

(5) Now run Migration Assistant but migrate only files, no settings, no applications, no extensions.

(6) Verify that bluetooth and email are ok.

If (6) fails, then I think you need to contact Apple Service.

It's important when you run Migration Assistant to migrate over user accounts that you want, and files but nothing else. No applications, no settings etc. This will bring over your email but you may have to re-enter passwords, server settings, etc.

By the way, if your Mac can go to Ventura, I would install that, it has better security and more bugs have been fixed, assuming you don't want to install applications that are incompatible with Ventura.

By migrating over only files, you will need to reinstall your applications, and in some cases re-enter serial numbers etc. But it gives the best chance to have a trouble free installation.

See the link Bob Harris provided. Sometimes resetting the SMC resolves an issue where hardware is locked up. Not sure if your issue with bluetooth is hardware. If it is, the reset may address it. It's a bit of a long shot. The erase/clean install eliminates all possibility of software causing the issues. Important to retest BEFORE migrating over your accounts and files, because if you still have those two problems with a "factory fresh" setup it has to be hardware.

You don't just revert to a previous OS.

You have to create media to install the OS of choice, wipe your drive (that destroys everything), install the OS, go through set up and then migrate data.

Part of the data migration is settings. If you migrate a corrupt or inappropriate settings, you're right back where you started.

It's not an evolution to be undertaken without lots of due diligence.

You might try an SMC reset. That takes about 30 seconds, the process varies according to the model.

Mac SMC Reset: https://support.apple.com/en-us/HT201295