Your battery can show 12.8v but still be bad. Hook it up to a known GOOD battery, and see what happens. From your description of the engine starting to lose power and then die, I would bet you dime for $ that a new battery will fix your problem.
PS. The flywheel doesn’t engage, the starter motor does. That’s another sign of a poor battery, it has enough power the spin the starter, but not enough to throw the bendix forward.