The battery can actually be disconnected once our OM617's are running.

The car may be stalling because the battery voltage is too low (at 12 volts) to heat the glow plugs adequately and/or spin the motor fast enough when cranking.
So.. yes, indirectly a bad alternator can cause stalling by affecting the battery charge.
You really need about 12.75 or more volts to start an OM617.
When the engine is running, a good alternator will be charging the battery at 13 or more volts.

Cold weather is hard on batteries, how old is the current battery?

Try cleaning the battery terminals and the cable connections and hooking the battery up to a charger overnight.

It should start right up if the battery accepts a full charge.

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78 W116 300SD 'Desert Rose' new as of 01/26/2014
79 W116 300SD 'Stormcloud' RIP 04/11/2022