Another possibility: Instead of a Mac Mini with external drives, a Synology NAS will also work as a Time Machine backup server. I've been doing that for a while and it's worked well. I do double Time Machine backups now: External USB3 drive when docked, and Synology NAS on the network. Basically, Time Machine alternates between the two.

I can't vouch for the cooling capacity of my unit, but it's been working well for quite a while with a couple of RAID drives in it. Earlier, I had a pair of dodgy drives but the RAID saved me there, and their replacement drives have been humming along just fine for quite a while, I think perhaps a couple years so far.