Odds of battery just dying?

Yup, seen batteries go out exactly like that and THEN recover to just like new after a few minutes of running! My theory is a loose/fracture cell connection on the grid. You can get "12.7 volts", but no ampacity to speak of.

A 3 yr. old battery is at the end of my comfort zone.

A great tool that makes those problems somewhat easier is a battery load tester, a relatively inexpensive must, imho. It thins out the guess work.