Any one of the uBlue projects is perfect for this use case.

KDE: https://getaurora.dev/ Gnome: https://projectbluefin.io/ Gaming: https://bazzite.gg/

Install and setup once, run forever. Immutable so impossible to break for a tech illiterate user, no package upgrades fuck-ups because updates are atomic and don't touch the currently running system, are done in the background and are completely invisible for the user, great hardware support, based on Fedora. Users can only install Flatpaks through the App Store.

The only "maintenance" needed is a weekly reboot to move to the latest OS image.

As a personal feedback, I moved my gadget enthusiast but tech illiterate father on Bluefin. He can ruin a Mac in less than a few months. He can generate undocumented bugs on iOS by his mere presence. But somehow, Bluefin is still running perfectly after a year. That's how robust it is.

Just go with Aurora (or Bluefin), that's the way to go for tech illiterates. Most of these recommendations are 5 years out of date, like why recommend Ubuntu or Zorin, when those are almost the same as Mint.

I’ve had several things break in the process of getting Mint installed and updated on this machine

Like what? That doesn't inspire confidence in the PC itself.

Debian Stable. Get it installed, get everything working right and configured the way this person likes it on a reasonable DE with default themes, and more likely than not, you won't have to touch this thing for years.

The setup's not necessarily for noobs, but if you're the one doing the setup, you should be able to get it into a place where it will pretty much never break for them.

You should probably give them KDE or GNOME (probably KDE, as it's more Windows-like and less my way or the highway than Gnome). As much as I love XFCE, it's probably a good idea to give a layman a feature-heavy DE so that nothing is likely to be missing; also, it's way too easy to accidentally delete panel items or entire panels on accident and a little annoying to restore things back to the way they were. KDE's panels implementation mitigates these issues.

Linux Mint

Since tech illiterate people don't really care what's under the hood as long as it works, I would choose anything with Cinnamon desktop: Gnome is a little alien for newcomers, KDE and XFCE are far too easy to screw up if you don't know what you're doing.
If Mint doesn't work well with you I would suggest either LMDE, stock Debian or the Fedora Cinnamon spin (in this order of priority)

I moved my wife's laptop to Debian with Cinnamon as a desktop. She loves it and is as technophobic a person as I know..

Auto login, automated-updates set up, remote backups. She just has to open the lid and firefox is there, which is 95% of what she wants. Libre office is around for the remaining 5%.

This is someone who used to get angry at Windows forced updates and reboots, so not having any of that improved her quality of life.

I would go for mint.

I want to suggest something immutable, but even when I use it, I have just had some issues occasionally or when trying to get the one off software here and there.

Of all the people in my family, even elderly, mint has been the easiest transition and I have very rarely needed to perform any additional maintenance outside of doing updates for them here and there.

Fedora kinoite for windows-minded users. Fedora silverblue for everyone else.

That’s what I’ve used for the old people I do tech work for and except for one who thinks Microsoft invented everything and nothing without MS branding is legit, they are all happy.

If you(they) don’t mind having outdated software Debian stable might be worth looking into. Otherwise there are immutable distros which are very hard to fuck up, and even if you do there is the option to rollback to the previous version. I’d recommend Aurora or Fedora (fedora doesn’t include some proprietary stuff like some codecs so if you need that it’s probably better to use Aurora).

Linux Mint also has a version based on Debian stable, LMDE, which is could also be an option. It’s not as stable as Debian as it adds its own stuff but has the out of the box experience.

As general advice I’d suggest using less packages and more flatpaks as a faulty flatpak update can only break that flatpak, not your system. For packages be sure to disable online updates, meaning you have to reboot to apply them. This isn’t as convenient but if stability is that important to you I’d go for it

Zorin OS

These days I just install Aurora Linux for this kind of situations. It's exactly how you described it. Haven't heard complains about it yet and their tech service needs from me almost dropped to zero.

Hannah Montana Linux

The typical recommendation in this case is Debian.
I don't know how hard it is to install since I don't use it myself. There are a lot of people swearing by it's stability, even running it for 10+ years on the same hardware without a reinstall.

People tend to reccomend the only one theyve tried lol, id say de matters more, most distros will just work in my experience, and most of them use guis for grabbing apps with the terminal being optional, Mint is an obvious pick, with cinnamon de being easy to use, prob wouldn't reccomend gnome with extensions unless you've already used linux for a few months, most other des have a solid default experience and easier (more straightforward) customizability

Bazzite Gnome- a hidden gem, as many think Bazzite is just for gaming, but it also has a great desktop mode. Pretty much indestructible (immutable), polished, pretty, has a modern kernel (so good driver support) and has Firefox and Libreoffice installable as Flatpaks. Great for kids and grandmas.

Ubuntu is a commercially supported os. That is my bar for recommending an OS to someone.

I put everyone on LMDE

FWIW, you can install Win11 on unsupported hardware with a pretty easy registry edit to bypass the TPM check. I did it for my mom's PC that lives 600 miles away from me, thus far with no issues.

If it has to be Linux, I'd recommend something that's immutable and atomic on a platform you're familiar with so as to prevent careless mistakes by technophobes.

Ubuntu Core, Fedora Silverblue, MicroOS from openSUSE are all pretty stable and have rollback if something gets borked. Unfortunately, nothing is guaranteed to 'just work'. Good luck!

But now with the end of Windows 10 looming, I need to upgrade a family member’s computer to Linux.

Why?

Did they ask for Linux? Do you have authority over them?

So this needs to be something that both is not going to break on its own (e.g. while doing automatic updates) and also won’t be accidentally broken by the users. ... There’s no way I’m going to be able to handle long-distance tech support if things break more than once in a blue moon.

Issues appear. I would be more focused on setting up remote access than choosing a distro.

I'd choose something LTS that has been around for a while (Debian, Ubuntu, RHEL-derivatives, SuSE if there's a freely-available LTS, etc.).

If you are not against the use of Google products, ChromeOS devices are about the best well-designed low maintenance operating systems. (Not Flex, a ChromeOS device.) But you would be sacrificing Firefox and LibreOffice, which might not be an option. (And technically, it's running a Linux kernel, if I remember correctly.)

Windows 10 will still be usable after support ends. Security is the only concern, and not that big of one if you regularly back up. It may be a good idea dual boot with Linux on a separate drive while you hammer out the issues with whatever distro.

I personally think anything with plasma 6 will fit for interface coming from windows. Some sort of distro like Ubuntu or Debian that commonly has pre-packages in the wild would make sense.

If Mint is misbehaving that badly on that hardware I'd be far more inclined to blame the hardware. What is it?

Maybe SpiralLinux? It's basically just Debian with a few twealks, but the btrfs+Snapper might come in handy if/when you need to do tech support.

All the normal folks I have moved over use Mint. It is not the distro I use. It has worked great for them. No complaints.