That sounds incredibly frustrating, especially after trying so many things. macOS permissions could be involved, but I would not assume that is the cause just yet.
The first step is to work out whether the scan is not being saved at all or is simply ending up somewhere unexpected.
Here are a few things worth checking:
1. Check whether the file was saved elsewhere. Open Finder and search for the scanned document's filename, if known. You can also check Recents and look for files created around the time of the scan.
2. Try a different save location. Create a folder called Scanner Test on your Desktop and scan a single page to it. If that fails, try another folder in your home directory. If you are saving to an external drive, network location or cloud-synced folder, test a local folder first.
3. Check the relevant macOS permissions. Go to System Settings > > Privacy & Security >> Files & Folders and check the permissions available for your scanning app. Only consider Full Disk Access if there is a specific reason to suspect it is needed.
Apple's guide explains how file and folder access is controlled: Control access to files and folders on Mac.
4. Establish which scanning app is failing. Are you using Apple's Image Capture or Epson ScanSmart/Epson Scan 2? They handle saving differently. If possible, test the same scanner with the other application. If one saves successfully and the other does not, that helps narrow down the cause.
Apple's instructions for scanning are here: Scan images or documents using a scanner and your Mac.
5. Check Epson's software compatibility. Confirm that your exact scanner model has software compatible with your installed macOS version. I would not reinstall drivers or reset permissions until this has been checked.
The important question is what happens when the scan finishes: does it complete normally but no file appears, ignore the folder you selected, open the scan without saving it, or display an error?
If you can share your Epson model, which scanning app you are using, and what happens at the end of a scan, someone may be able to pinpoint the problem without making you repeat all the troubleshooting you've already done.