Hello again,
In the light of the October debacle, I was under the impression that all signing/certificate problems had been solved by the two companies...
It is interesting that the LaserJet 1012 still today is mentioned in an HP article https://support.hp.com/us-en/document/c05900790. In this article, there is a link for driver downloads:
Please note that that link has another address (ftp://ftp.hp.com/pub/softlib/software12/HP_Quick_Start/osx/Applications/ASU/HPPrinterDrivers5.1.dmg). You may want to check whether this download would change anything.
Also, there is a note in the HP article https://support.hp.com/us-en/document/c06960563:
"This might be HP Easy Start, HP Smart, or Mac Printer Driver version
5.1. System administrators and those unable to use HP Easy Start can use
HP Easy Admin to download the software pkg files directly for use in managed or remote deployment environments."
You may want to investigate whether that would allow direct access to a working LaserJet 1012 driver download.
If everything else fails, a LaserJet 1022 driver may or may not work with the 1012 printer. The LaserJet 1022 seems to be supported for macOS 11 Big Sur (at least according to the HP download page for that printer). A discussion Driver HP LaserJet 1020 MacOS Big Sur - Apple Community about using a LaserJet 1020 with a 1022 driver could perhaps also be of some interest in this context.
Anyway, good luck!