Time Machine stuck "Preparing backup," out of obvious fixes

No Apple store nearby (in fact, I have been this community's Apple Certified Macintosh Technician for 20 years) and really can't afford the shipping wait.

Yeah, I'm aware of the heat issue with kernel_task -- it hits my MBP every time ambient goes over 80°, and I used to use it on outside jobsites in AZ a lot. But that hits all the cores, plus I've never seen it on the Mini.

After restarting after removing the nonexistent folder, kernel_task seems to have stopped spinning. Yay!

Again, it's stuck at "Preparing backup." Now I am seeing the following errors in the log (I've seen them before, too). This whole "file:///" issue may be the key to the problem. I have no idea what it's trying to do here. There is a file .com.apple.timemachine.supported at the root of Tardis, but it's zero-length, and other discussions indicate it should be, so I don't think it's involved. There's nothing else in the hierarchy that looks like it would be involved in whatever TM is trying and failing to fetch.

I ran DU First Aid on the TM drive back in the OP. Since I had completely removed the Backups.backupdb folder in order to start fresh, there was no multilinked anything remaining on the drive and the operation was quite prompt. On that drive there are only three sparsebundles of backups from three other household machines, plus a bunch of invisible cruft belonging to the system. The backup drive is in Spotlight privacy.

My boot drive is 130/500GB full, a pittance in today's terms. Of that, /Users takes under 13GB, and /Developer runs 2.5GB, the rest is all vanilla macOS. However, your question reminded me that there is also an external drive for large files, some of which does get backed up as well, and I had never yet checked that. DU loved it long time, but DiskWarrior said it was replacing the directory because the old one was too damaged to compare (no data was lost). For kicks, I also ran DW over the backup drive (since there is no multilinking on it yet) -- DW reported trivial corrections similar to those reported almost any time you run it.

I restarted TM, got the same spate of "file:///" errors, and it's been "Preparing backup" for about five minutes now. I'll let it run and see if that fixes it, I'm thinking my odds are only maybe 20%.

I've run DU First Aid on every volume on that system at least once in the past 24 hours, and DW on the two externals (the internal is APFS). They are now all as clean as they can be.

Both externals are RAID-1 (mirror) volumes that well predate this Mini; I expect the archival volume structures collected some damage during power outages or whatever, or maybe even due to some bug in a version of the RAID driver (SoftRAID). I'm still a bit fuzzled about how DU could say it was perfect, while DW said it was beyond use -- what kind of damage does that? In any event, I'm pretty confident the hardware is solid.

I have no reason to even suspect anything is wrong with any of my hardware. Startup-D gives a clean bill of health, and there's really no reason for the SSD to be going sour. (And if it happens again, I can just replace it, once I have a recent TM backup).

I am currently using another (working) A2 device to clone my drive from the Mini to an external so I will have something to run File Migration from, then I am going to nuke the internal and reinstall a fresh Mojave.

Yes, you bet I am fixated on APFS. When I get loads of error screens from (useless) DU saying stuff like:

error: Cross Check: External physical extend (xxx + 1) has kind APFS_KIND_UPDATE 
      but was not referenced previously {several of these}
Snapshot is invalid
APFS inverter failed to invert the volume -- invalid argument

...I don't have to run DNA to ID the probable culprit. I repaired and maintained Macs for 35 years without ever having to "invert a volume," whatever the heck that is, and it looks like something I could easily have gone without.

If DU were up to its job, Alsoft wouldn't be in business. But there's a reason they've been profitable for decades.