I wonder if the author is right; using a slightly wrong capacitor for the clock (keeping in mind that they're usually +/- 10% or worse from the factory) may produce a different oscillation speed, but usually not an outright failure. These chips accept a pretty wide range of clock frequencies.

And in the comments, they note:

"I replaced the 18pF capacitors on one of the non-working boards with 10pF capacitors, but it still doesn’t boot or respond to the debugger. That surprised me – I thought that was the answer! It was a rush rework job and I made a bit of a mess of it, including accidentally desoldering and reinstalling the crystal, so I’ll try again later with another board. But it appears that the capacitor value may not have been the issue after all. Either 10pF is a bad value, or it’s the crystal itself that’s at fault, or I’ve failed somewhere in my troubleshooting reasoning. Hmm."

In my experience, mystery stability problems are often caused by capacitors, but of a different kind: decoupling capacitors on the power supply pins. If there's not enough of them to keep up with the noise originating from motors or digital switching, I'd expect that exact issue. Intermittent "impossible" CPU states on some boards, no rhyme or reason (because sometimes, that +/- 10% saves you and sometimes it does not). I'd try more caps and possibly some ferrite beads.

I'm really surprised that nowhere in the article does the author mention putting an oscilloscope on either the power rails or the clock signal. Normally when I'm troubleshooting broken devices those are the first things I check.

A multimeter can make it look like a chip is receiving the correct voltages, but there can be all sorts of noise and fluctuations that will cause really strange and inconsistent issues. You don't need a super fancy scope, even a 100mHz cheap handheld is more than good enough for checking power rails.

Also, for anyone who has to troubleshoot or fix boards, those smd resistor and capacitor kits are absolutely invaluable to have on the bench. After a decent scope, meter, and soldering station I would say that should be the next purchase for setting up an electronics lab. Nothing more annoying than trying to debug an issue and having to wait for the right value cap to be shipped. Especially since capacitors specifically often seem to have some trial and error to finding the right value for something like a decoupling cap. Fully modeling the noise generated on a supply rail often isn't completely possible, at least in my experience.