System Keychains no longer accessible and have an unfamiliar icon

Try opening the network keychain via File -> Add Keychain. When I created a new keychain it appears in the sidebar under Custom Keychains but your screenshot shows System Keychains. You might need to remove the keychain first then add it back. This won't delete the keychain itself just disconnect it from the Keychain Access app. Make sure you choose the option Remove Reference and not Delete Keychain.

You could also create a new keychain and then use the Import Items option and import to the new keychain via the Destination Keychain drop down list.

The keychains should be stored in the following location:

~/Library/Keychains/

or

/Library/Keychains/

Make sure the files exist such as "network.keychain.db" and if the permissions are correct. Check your backups as well from back when they were working. You'll need to remember what the passwords were, etc. to unlock them.

If they are located in /Library/Keychains that would be the System location and personal keychains should be located in each users ~/Library/Keychains location and you will need to be the owner and have read/write permissions.

Sounds like the files were an older format as well. I only have the filename.keychain-db files and not the secondary but I do remember that being the case in the past. Did you skip macOS versions upgrading to a newer macOS version? There might be a way to fix this with the "security" command in the Terminal or manually removing the old files and reboot but it might not be worth the effort if things are working fine and the only issue is the bad system keychain entries that remain.

Make sure you have a good backup before you mess with it.