Restoring previous OS (Ventura) from Time Machine backup after a repair

It’s likely impossible, yes. You would not be able to restore from your backup. While you could consider manually copying off your data then manually restoring it, knowing you’d lose settings, licenses, etc., jumping 3 major versions as your Mac did almost certainly involved firmware updates that would preclude proper functioning under the much older OS. That means you’d need another Mac and end up using DFU mode and the Configurator to restore the firmware, etc.

Bottom line is that it’s technically potentially possible, but practically you’re better off adjusting to Tahoe. Many major software platforms (MS365, Adobe, etc.) only support the three most recent versions so they won’t support Ventura anymore now anyway.

If there is a backup from when you had Ventura, you can migrate from that time after you install Ventura. When you enter migration and pick a backup, you need to pick the backup from the last Ventura backup.

When you use Migration Assistant, you can select a backup from those available on the TM disk. There will be a list of the backups on the disk:

Note that you would need to boot into the Recovery OS and completely erase your Mac's internal storage before installing Ventura.

I'd still suggest being prepared to follow these steps:

How to revive or restore Mac firmware - Apple Support

There are certainly some adjustments to be made, but if there is something you feel is not working properly you could certainly post that as a new question and hopefully help can be provided for a specific issues.