I had exactly the same issue with my HP Deskjet 3070A. As suggested I installed HP Smart and I was asked to create an account. I did not want to create an account to simply scan a document.
I found an alternative. You can install HP Easy Scan:
https://apps.apple.com/fr/app/hp-easy-scan/id967004861?mt=12
This program installs the scanner driver your Mac does not have.
At some point, you will be asked if you want to install HP Smart because it is more recent. You seem to do not have alternative but you actually do. Try to quit the installation window then you will be asked to:
- continue with HP Smart install
- continue with the current HP Easy Scan installation
- cancel your exit action
Select the second option.
When asked, you need to install only the "Essential" part.
Once installed you can safely quit the installation without providing personal information such as your zipcode.
At this point you can remove your printer from "Printers and scanners" window then re-add your printer using AirPrint. This is exactly what you did before but now the "Scan" will appear.
Unfortunately your mission does not stop here. When trying to scan something, you probably will receive an alert (as I did) mention you do not have the permission to open the application "HP Scanner 3". This time the HP Support got you covered with this help page: https://support.hp.com/us-en/document/ish_4448469-4448558-16
I hope this message will help you.