Connect to wifi after restore

I think try another network to see if the issue persists.

The other thing is just restore it as a new mini, without restoring from your backup and connecting to the network, if possible. Then you can try wi-fi connection once its up and restoring the backup from there.

Otherwise all I can suggest is you will need to take it in to an Apple store for testing.

Maybe the router is rejecting new connections? Routers have a setting to permit or reject new devices.

But its possible the mini is confused, did you try powering it off and on and restarting the restore procedure?

Does it display an error message?