Your point about "issues" with some things on the Time Machine backup is well stated. I've often wondered about the stuff on backups, and how stable/safe/etc. it is. I use a different way of backing up, as SuperDuper! just makes an exact, bootable clone of my system (that is the case for both of my Macs). I certainly put a lot of effort into keeping my machines "lean, mean, and clean", but I of course cannot be perfect. All I can do is try my best. Each of those bootable clones, though, work fine on their own, that is, I can boot each of my Macs from either clone, and it works fine (as far as I can tell). I actually just boot from the machine's clone. That is, I boot my Mac Mini from its clone, and my MacBook Air from its clone.
Recently, for a couple of the newer releases of Sierra, I actually booted to the clone, used Disk Utility there to Erase and Format the internal drive (SSDs for both of my Macs), did a fresh, clean, "virgin" installation of the new version of Sierra, and then used Migration Assistant to "migrate"/copy needed "stuff" from that SuperDuper! backup. I then re-booted my machine, and everything was fine. For the other ones (like the recent OS 10.12.5 update, I just downloaded and applied the OS 10.12.5 Combo Updater.
In actuality, it would be faster for me to just recover directly from the SuperDuper! backup. In fact, it is very straight forward, and when I done that, it worked well. The only possible "glitch" is that SuperDuper! does not backup the (hidden) Recovery HD partition created by an installation of the Mac OS (there are other ways of creating it), but in reality, I 1) rarely, if ever, boot to that partition, 2) can get to Disk Utility in an "isolated" manner by booting the respective Mac from the SuperDuper! backup, 3) use TechTool Pro there to perform disk cleanup, maintenance, and repairs (does a better job than Disk Utility), and 4) can do a fresh, clean, "virgin" installation of the Mac OS I want (I always keep two copies of the applicable "Install macOS Sierra" file, and they are of course on the SuperDuper! backup).
Your issue is quite possibly more tough to pin down. It could be an "obscure" hardware error as you conjectured earlier. But as long as your internal drive checks out as OK, taking it out of your Mac, erasing and formatting it, and then doing a fresh, clean, "virgin" installation of the Mac OS you want is wise. Now, I do not use iCloud, as I have all my applications stored in a folder entitled "Useful Applications" (this is the case for both of my machines). And of course it gets backed up by SuperDuper!. How "safe" is iCloud for storing your apps? If you know it is safe enough, then maybe doing a one by one installation of each third party (I assume you have the correct version for the Mac OS you are using), though a burden, would be the best way to insure that things are good, at least from a software perspective. And of course when that is all done, you can actually boot your Mac from that "populated" hard drive inside the enclosure.
As a backup, do you have your third party apps stored someplace on your hard drive, in addition to iCloud? That would not be a bad idea.