It is interesting that HP's own app does not detect an HP printer...
Also, the difference between macOS 11.2 and 11.4 should not explain why it is not working.
The HP support page only suggests HP Easy Start (https://support.hp.com/us-en/drivers/selfservice/hp-ink-tank-310-series/16180943/model/23086544) for this printer model (and 11.2). There have been some reports about difficulties with that app for other models.
According to the specifications (https://support.hp.com/us-en/document/c06133051), the printer in question uses HP PCL3 GUI (this is not standard PCL). If absolutely nothing else works, you could perhaps try drivers for other HP printers using the same printing language. That would be experimental, though. As with any experiment, things may go wrong. It may not work at all. Software/hardware problems cannot be ruled out. Data may be lost. So, do not try anything unless you know what you are doing. In any case, make sure that you have at least three backup copies of important files (photos, et cetera). The article https://www.ibm.com/support/pages/information-printers-hp has some information about older printers using an HP PCL3 GUI data stream. The (Dec 11, 2020) HP Printer Drivers v5.1 for macOS from Apple (https://support.apple.com/kb/DL1888?viewlocale=en_US&locale=en_US) may be of interest as well.