Is there any systematic strategy for keeping apps from PPAs up-to-date? Or do you just have to keep a manual list of what you've installed using PPAs?
PPAs are, from apt's perspective, just perfectly normal repositories from which packages (and updates) are fetched. You don't have to do anything special.
But only apps which are being "package-managed", as I understand it.
right! And that includes PPA-installed packages.
Today I also upgraded an app in another way: apt had installed restic on my system, but only version 0.16. The latest version is 0.18. I was told how to get hold of this by the devs in the restic forum: it involved downloading a tar.gz from the corresponding github "releases" page, extracting the file, making it executable and putting it somewhere appropriate.
Well, yeah, but that's not great for automatic updating, as you noticed. A different way would have been to download the Mint (meaning: debian) package sources. Maybe they are already at version 0.18; if not, you could have updated the build recipe form 0.16 to 0.18; build the apt package and installed it via apt. That way, it would have automatically updated when mint included 0.19.
But with apps which don't, but which aren't even from a PPA, what's that recommended way of "remembering" to upgrade apps at a reasonable frequency? Just keep a list?
Yes.
Is there any reason why versions in the apt package management system are seemingly so laggy?
You're using Mint, which is intentionally very conservative (it derives from Ubuntu long-term-support releases, and these themselves are typically behind debian testing); so, this is a matter of choice of Linux distro. For example, debian stable is shipping restic 0.18, Ubuntu 25.04 is shipping restic 0.17, Ubuntu 25.10 ships restic 0.18. Mint also doesn't do very frequent distro releases, so you cannot just "hop on a newer Mint version" very often.
But: if you like Mint, and want the extra speed, maybe try Ubuntu? It does put out an update every 6 months, and technologically, it's pretty close (and all the apt commands you've learned work the same).
If you like bleeding edge software (I'm not sure I would want to use bleeding edge backup software? I'd be pretty happy with someone trying it before me… then again, I'm using a very recent version, too), then maybe go for either debian unstable (which contains "software versions that have been tested thoroughly but could not end up in debian stable yet, because they're a major change"); if you want to try a fast-moving regular-release distro that's not using apt but dnf, I do like fedora. But then you need to learn a new set of package management commands (they're not completely different. You install, uninstall, update, and search for packages).