I've seen the videos, but a answer hasn't been forthcoming.

That is, if I start with Timeshift in LinuxMint, then say (for example) a year later I switch to Manjaro, a completely different animal, and on day 2 of being on Manjero, I needed to restore my system, could I use the files Timeshift created on LinuxMint to restore a system running Manjaro at a later date? (or other distro, than the backups were created on?

Thanks...

My experience:
Let's say your latest Mint backup is from July 31 and your first Manjaro backup is from August 1
It 's August 2 and you do a restore from the August 1 backup = Manjaro. If you do a restore from July, you switch back to Mint and you loose Manjaro.
So you can surely restore one distro with another. I have done this very often (20 or 30 times or so).

AFAIK this is by design of Timeshift.

Disclaimer 1: I use MX Linux and Debian. They are pretty close.
Disclaimer 2: I noticed some times slight differences between a backup over the existing FS and an emptied FS. I never found the cause (never did a serious root cause analysis, but suspect something with timestamps of files), since I mount my FS's with noatime parameter). A backup to an emptied FS was always OK (checked with the MD5's, as well as the behaviour). So test this, to be sure. If someone has an answer to this... I'm curious!
My solution: I always use a triple boot system (dual boot should do), with a distro that includes Timeshift, to be able to restore to am emptied FS. A distro on a bootable USB should work as well of course. this way I know I do not mix things up.

I use Clonezilla with none of the concerns OP mentioned.
AFAIK, Arch-based has one version, 'buntus and Mint have another; Maybe Debian has its own (@remmilou); I doubt Slackware users have much use for Timeshift.

Yep. I use Clonezilla as well. Absolutely great and never let me down. Probably your safest bet, when you change distro's. The downside is that it is more work. Timeshift is "set and forget".

No.
Backups are not some sort of magical sauce you can apply to any system.
You shouldn't be asking "can $SOFTWARE do this", you should be asking "will my backups be compatible with a completely different system".
Also, backups of what? The whole system? Probably not, because your question wouldn't even make sense. Backups of selected configuration files? We need more information about your backups.