Ok, I've just spent a significant amount of time over the past two days trying to fix this issue. No success, but have some interesting information that could be helpful.
1)
Setting up a separate user account does not solve the problem on the second user account. Therefore problematic files are not in the user directory.
Setting up a separate partition with a blank installation of macOS does fix the issue on that partition, however as soon as you restore from your backup (e.g. Time Machine), the backed-up Apple Pay preference files are written over the working ones and the problem will occur.
2)
I have had a play around with the following files, with varying degrees of success.
You will need to enable hidden files. Some files require you to add yourself to the permissions to read/write them. Some require system integrity protection to be turned off to edit.
Replacing the /System/Library/PreferencePanes/Wallet.prefPane file with a fresh copy from the new macOS installation does not help. In fact at one point it stopped it working entirely.
I tried deleting the card files (these are stored in /Users/~/Library/Passes), this didn't seem to help either.
I also tried replacing them with the files from the fresh-install partition. This also didn't work.
There are some Apple Pay files in the following protected & hidden folder (/private/var/db/applepay/Library/Preferences).
If you delete com.apple.seld.plist, then restart, Apple Pay then allows you to reset and add a card. The problem then returns next time you restart, when you are again unable to add any further cards without resetting Apple Pay. I also noted that deleting this file stopped the card I did manage to add working on Safari (I tried to buy something from the Apple Store with Apple Pay, and it just didn't come up when I clicked the Apple Pay button). The file isn't re-generated by the computer at any point after deleting. Perhaps this is the problem file that needs to be regenerated after a restore?
I tried replacing all the replacing all the files I could find with the fresh ones from the working macOS partition, this simply generated error messages when trying to load the Apple Pay preference pane.
I get the feeling I'm missing some files!
I think apple needs to introduce a new Reset Apple Pay button, which completely restores all of the files associated with Apple Pay (and the secure element). Then after restoring from a backup it can be reset.
This surely will be a big problem as more people experience issues with their machines (e.g. the logic board failure I had), and are required to restore from a backup.
PLEASE HELP APPLE!