Which coolant sensor. One provides input for temp gauge. There is another that provides input for engine management system. Depending on the engine type, fuel injection system, etc, the loop feedback from the sensor will enrich the mixture when cold starting. Will also affect timing in some systems. Shouldn't play a significan role in a hot start failure scenario. Maybe Benzmac can provide more detail.

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Jeff L
1987 300e
1989 300e
1987 BMW 325