Your first assumption was correct!
The charging system is the problem. The "technician" that said the car couldn't kill the battery that quickly is mistaken.
That model & most other new cars can consume as much & more power than the alternator can generate.

You need 120A and more output from the alternator to keep up with the demand.

Have you checked with your MB dealer to see if that ML falls into the voltage regulator recall that affects 350,000 2002-2005 vehicles?

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