I used to use my late 2014 Mac mini at home with an external hard drive connected as a shared drive to backup several Macs using TimeMachine. This has worked well for years now, but recently the clients stopped to recognise the Mac mini as TimeMachine server. I followed these two instructions and now it works:

1. The official Apple support post to set up your Mac as a TimeMachine server (essentially: right click on the shared drive in the Sharing preference pane and select it to serve as a TimeMachine disk). This made the disk appear again in the client’s TimeMachine preference pane, but it wouldn’t allow me to connect saying (translating from German) “You don’t have the required permissions to use this disk”. Only after following the next step, did it work.
2. This user-contributed instruction on the Apple discussions forum (essentially: create a folder on the shared disk and share the folder rather than the whole disk)

Don’t bother with all the old tips on the web saying that you need to download the ca. 30 € Apple server software. It doesn’t support TimeMachine anymore for newer macOS versions and it is not required anyway, as you can also just make it work using the built-in sharing feature.