Make sure the housing of the alternator is grounded by a wire to the engine block. The engine itself is grounded back to the battery on the other side.

At 12.4 volts the battery is only at half a charge but the charging voltage should hols at 13.2 to 14.2 most of the time when fully charged battery and running with a variety of loads.

A battery on its way out will not rebound all the way back up to 12.6 after some light usage. By putting a voltmeter leads down into the acid of each cell, to cell, you can find the weakest one.

Adding a voltmeter to the dash panel is a good idea, so you can watch for any changes in the future. You can set up a mental set point for what is normal after starting and after the battery charge stablizies.

What did the slip rings that the brushes run on look like? We're they smooth and not worn very deep? Maybe, I should ask, how many miles do you think is on it or is it a rebuilt unit?

Phil