Should I upgrade macOS on this older MacBook Pro?

I had the same experience. Some of the Apple links for older systems only worked if I was using Safari.

There is apparently an option for fixing the older installers that was supplied by Randy Singer. Scroll down until you see it:

Like you, I’m working hard to keep my mid-2012 MBP working because of the software I’ve invested in won’t work on the most current systems.