K-jet system?

I'm not extremely familiar with the K-jet in the Benz, but have had quite similar systems in SAABs in the past. Yours may be very similar.

There are two independent enrichment mechanisms for starting, and you need them both.

One is an additional injector in the intake manifold plenum, which squirts a bit of fuel while you are cranking the engine. It's limited by the "thermo-time switch" in the coolant. That device allows the injector to run if the coolant temperature is low enough, but it heats itself as it passes current, eventually cutting the flow. Failure of anything here will tend to just affect starting, and not driving off even with a cold engine.

The other is the warm-up regulator. This sets the fuel consumption based on the temperature at the water outlet, by means of a pressure regulator in the unit. This pressure forces against the airflow plate, so more pressure means less fuel. Maximum flow through the warm-up regulator causes low control pressure and high fuel delivery to the engine. If the regulator is plugged with debris, it will not properly enrich mixture for a cold engine. If this is the case, your engine won't run quite right when it's cold.

There's also a lambda feedback enrichment device, which diverts fuel from the warm-up regulator, but this shouldn't be an issue when starting the car.