The HP Printer Driver Package 5.1 does not include and never did include drivers for your HP Officejet 8710, that is why you are not being offered it, so stop looking.
Have you downloaded the updates available to you from the HP page in the link above, it does not matter if they
were released before the certificate expired. The firmware update might be important and allow the printer to download
any new software when resetting the printing system. If you click on the link to HP again and click on the + next to driver
you will see HP have released a new version of HP Easy Start on November the 2nd, downloading, installing and running
that should search for any updated software, firmware etc.
Are you resetting the printing system correctly.
Open System Preferences> Printers & Scanners.
Control-click while the pointer is in the left hand panel, Reset Printing System should pop up, click on that,
you will be asked if you are sure you want to reset the printing system, press Reset.
Resetting will take place, it will take a minute or two.
Make sure your mac is connected to the internet, and your printer connected by cable to your mac
and your mac connected by cable to your router.
Click the + button in the left panel and add your printer back in.
At Default Printer select HP Officejet Pro 8710, this should force a search for any new software etc.
I also notice that your printer is an AirPrint printer, which means it should print over WiFi without
the need for manufacturer drivers.
Read this article from Apple.
About AirPrint – Apple Support
Use AirPrint on your Mac - Apple Support
What have HP said about your issue.
