A running engine on a disconnected battery does NOT guarantee a broken alternator.
A had it happen to me. Car came to traffic lights and died, and the lights went out..a seconds thought revealed the most likely problem was a loose battery terminal. It had in fact shaken right off over a bump.
Putting it back on and bashing it with a boot heel restored normality.
There are ways to blow and alternator,, but total loss of a battery to charge is not one of them.
A shorted battery is a different matter.