My 27" iMac 5K (2019) has become almost unusable

I bought a new iMac last year; here are the stats:

• Currently running Mojave 10.14.6; plans are to upgrade to Catalina as soon as i figure out the current issue I'm posting about
• Upgraded memory to 128GB
• 3 TB Fusion Drive is little less than ⅔︎ full
• Making backups via Time Machine to an external 4TB hard disk… which i'm just noticing is having it's own problems (currently showing as "Untitled" generic USB Disk in the About This Mac window, even though it should be showing up as Deep Space 9.

Showing up in Disk Utility as disk5s2 and is unmounted. Clicking Mount doesn't appear to do anything.

My mid-2015 Retina 2.8 i7 MBP had become almost unusable because I had left it plugged in all the time and the battery had swollen to the point where it was pressing against the bottom of the trackpad and keyboard, making mousing and typing erratic and unpredictable. Oops.

Anyway, I have this nice new iMac that has ended up being slower and less reliable than the MBP.

As I recall, I used the Migration Utility to transfer the apps, documents, and system files from the MBP to the new iMac. That might have been my first mistake, since I hadn't done a clean install of system software in years and years, so who knows what detritus from macOS systems from long ago were still present in the data I transferred.

Symptoms:

• Everything, from simple Finder tasks (copying files) to launching apps is p a i n f u l l y s l o w.
• Mail crashes almost as soon as I launch it. (Mail runs fine on the MBP and on my various iOS/iPadOS devices.)
• Preview: same thing. If it manages to launch, it almost always crashes immediately.
• The Finder seems to have trouble updating meta data about files. Or negotiating the difference between iCloud files vs. local files.
• Spotlight sometimes cannot find files that are, indeed, on the disk I'm searching.

What I've already done:

• Run Disk Utility ➔ Disk First Aid (a lot).
• Forced a relaunch of the Finder far too many times, which I (think I) know can introduce corruption into file structures. My bad, but one can only look at the SCWOD (spinning color wheel of death) so long before one takes matters into one's own hands.
• In fact, because of the unreliable nature of launching apps, I have begun to frequently resort to a Force Quit and relaunch. Probably Not A Good Thing.

Ultimately, I know I need to do a clean install of Mojave, restore from a backup (which may be thwarted by the weirdness I've described above…), and finally upgrade to Catalina. Part of the challenge here is that I have a fairly "customized" system: I've been using Macs since the first one (1984). I run with a bunch of preferences, login items, and RAM-resident utilities that may be slowing things down and/or causing/exacerbating some of the issues I'm complaining about. I know that Catalina will force me to abandon some of these because they're not compatible with the new 64-bit architecture. That being said, I run the same things on my MBP with no problems.