HP Support – HP Deskjet 3050 All-in-One Printer – J610a
HP's specifications state that it "Supports 32-bit only Mac OS X v 10.5 or v 10.6: PowerPC G4, G5 or Intel Core processor." And at first, I did not see any Mac drivers for this printer on HP's site. Later, when I clicked on the "Change preferred operating system" link, the HP site listed four ancient versions of Mac OS X: 10.5 (Leopard) through 10.8 (Mountain Lion). (Mountain Lion came out in 2012 and is twelve major versions older than Tahoe.)
I wouldn't expect any of those old drivers to work on a Mac running Tahoe, whether Intel-based or Apple-Silicon-based. You say the Mac wouldn't allow the installation of the drivers, which suggests that it saw that the drivers are Intel-only and possibly also 32-bit only and rejected them on one or both of those grounds.
The printer does not appear to support AirPrint – which would let you print without a driver. HP's specifications say "Mobile Print capability not supported", and the printer is not listed in Apple's database of printers known to be AirPrint-compatible (About AirPrint - Apple Support).
It looks to me like you are going to be in the market for a new printer.