Hi,
You have mentioned more or less all options that one would investigate regarding drivers for old HP printers.
The supported printer drivers from HP's support page for the HP DeskJet 990Cxi are for Mac OS X 10.4 Tiger and earlier.
A Gutenprint driver exists for an HP DeskJet 990C. This may or may not work with macOS 12.6 Monterey. Also, the normal recommendation is to download from the printer manufacturer or from Apple. Downloads from other sources would be at your own risk.
For the last two packages, the question remains whether or not they will work with macOS 12.6 Monterey. If you could install them, they may contain drivers at least for similar printers. The DeskJet 990Cxi uses PCL3 enhanced (PCL3e), so a driver for another model with the same language could perhaps be tested. However, all this would be entirely experimental. As always, make sure that you have backup copies of all important files before testing anything.
If nothing else works, you may have to look for an older Mac model, running a compatible operating system version, to be used for the connection to this printer.
http://h10032.www1.hp.com/ctg/Manual/bpd08297.pdf
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
Reset the printing system on your Mac to solve a problem – Apple Support (UK)