You do not use macOS "Sharing" preferences to create a shared folder on the NAS. The macOS "Sharing" System Preferences is only meant to allow items on your macOS computer to be shared out to other users and/or devices on the network.
You need to create a folder on the NAS itself and configure the NAS to allow for TM backups. Consult the Synology User Guide on how to configure the NAS to work with TM. Once you have the Synology properly configured to accept TM backups, then you just need to use the TM app on macOS to point to the network share on the NAS as the backup destination.
Edit: You should never upgrade to a completely new OS to solve a problem unless you have been told by the hardware vendor or software developer that doing so will definitely solve a known issue. A brand new OS is usually likely to have all sorts of compatibility issues especially within the first few months after release. Upgrading to a new OS (especially a recently released OS) will usually just make things more difficult & complicated. It is best to sort things out on the current OS first.