There's another box in the charge circuit, the overvoltage relay. It
could be tripping at too low a level. And if the battery is old or has
been overcharged, it won't charge regardless.

Put a voltmeter on the battery terminals, check the voltage. Then
start the engine, turn on the alternator switch and check it again.
With the engine at taxi speed, you should get over 13 volts if the
regulator circuit and alternator are working. If you're getting 13
volts and the battery still dies, replace the battery.