Whoever you talked to at HP support should be fired. FWIW, I have been using HP printers (lasers & inkjets) with my various Macs for years without issue ... unless you consider running out of ink an issue. ;)
To make sure I understand your statement: "I have a new router now and am unable to connect it," do you mean:
- You are unable to setup and connect this router to the Internet? or
- You are unable to setup and connect your HP printer to this router?
If it's the latter, then let's start with what exactly doesn't work with your printer:
- It won't connect to your router's wireless network?
- It connects but you are unable to print to it from your Mac?
Your printer is Apple AirPrint capable. That means it doesn't need to have you check for updated printer drivers. You should be able to print to it, via AirPrint (wireless), from your Mac and iOS/iPadOS devices.
Ref: