Downgrading macOS versions requires wiping the drive before installing an older version, as far as I know. Trying to run the installer for an older macOS version directly won't let you install. I won't say there aren't other ways, but at least wiping the drive and installing is well-known to work.
The steps look like this:
1.
BACK UP YOUR FILES!! YOU DELETE EVERYTHING FROM YOUR COMPUTER IN THE UPCOMING STEPS!! Using Time Machine is a very good way. Other third-party tools like Carbon Copy Cloner are nice too. Honestly, if you have the time and experience, you can grab files by hand too onto an external drive or cloud storage. Manual backups are tricky since there are a bunch of hidden metadata files you might want.
2. Create a bootable installer:
https://support.apple.com/en-us/101578
3. Boot into that USB device: restart your Mac and hold the Option key to select the USB as the boot drive. All your files and system are still intact at this point. (This step might vary between Intel and M-series since they reworked this stuff between generations)
4. THIS IS THE POINT OF NO RETURN!!: while in the USB's installer interface, use Disk Utility to erase your startup disk (selecting the Macintosh HD volume and clicking Erase)
5. Proceed with the macOS installer to reinstall the older OS
6. Use your backups to restore files. For Time Machine backups, attempt to follow:
https://support.apple.com/guide/mac-help/restore-files-mh11422/mac
Step 6 can go in various paths, from restoring specific files at a time, to trying to do a full restore of everything.
That may require using a Time Machine backup taken on the older macOS version. I'm like 60% confident you can restore a TM backup from Tahoe onto earlier versions, but YMMV. Even if you can't do a full easy restore from a Tahoe backup, you 100% can grab all the files it backed up from the drive and put them back. It just won't be as easy and depending on whether it's metadata you're trying to restore, it gets exponentially more difficult.
Personally, I find it easiest to attach the Time Machine backup drive, navigate to a timestamp folder for the backup files in Finder, and drag over specific files. This is the least assisted way of doing things, but it allows me to have the cleanest start for the reinstall. Keep the backup drive and files around, and you can always go back and recover more files that you need/forgot.