For my old HP2100 (using a network connection) I usually just use the gutenprint driver that is a choice when you set up this model. I found it was way faster to print PDFs. This is not affected by Big Sur.
To use the HP driver I used the uninstall HP software and then reinstall HP’s 5.1 driver from Apple support using the shareware Pacifist [suggested in an earlier posting as a workaround] I would still get the frightening warning dialogue box when printing but I just ignore it in order to print. I see now the 5.1 is available and hope it has been updated so it can be installed on its own and no more warning boxes! I’ll try it later.