My journey to figure it out found me a Monterey IPSW image. Try to install it via DFU and a second Mac. "Nah, you can't do that, I won't even let you try."
ChatGPT hinted that I needed to do it from a similar vintage OS. "Even an Intel Mac running Ventura could work for this." As luck would have it, my partner still had her old MBP Core i5 running Ventura!
Alright, install Apple Configurator on the Ventura Mac.
"Nah. You need a Mac running 15.7 to install Apple Configurator."
Chicken and egg.
I mean, this OS (Monterey) only came out FOUR YEARS AGO. Ventura was three.
I got lucky with a Reddit post where someone asked for and got a zip file of an old version of Configurator.
I was then able to DFU re-image the M1 Mac with Monterey.
(Why do I need Monterey on it? Because someone else abandoned their software.)
So this Kafkaesque process to even get a four year old OS on a four year old Mac laptop means we shouldn't just be slobberingly praising Apple.
(I realize you, personally, weren't. Just when you said 10.12, I got flashbacks.)