Just for piece of mind let me tell you how your warning light works.
When the key is turned on power goes through the light to one of the brushes in the alternator. It proceeds through the rotating field through the other brush to ground.
The alternator acts as a ground and the light lights. Once the engine runs and the alternator spins this small current energizes the charging process. There is from that time on power from feed diodes to the same brush. This causes the light to have power on both sides so it no longer lights.
It only takes the potential to charge to rid the light. It makes no judgement on capability. Unfortunately its a very poor determinant of the charging systems ability to KEEP up. Once the battery gets below 9v the control units start shutting down.
This is a remnant of simple systems and surely will be replaced with the coming 40v systems. It would be nice if the car manufacturers could put in system status warnings, I'm sure they will.
Bosch Master, ASE Master, L1
33 years MB technician