The problem with that reasoning is a), you also likely have an inbound missile to defend, and b) IRL you defend based on MAR, not whether you have a missile launch warning, as you cannot depend on missiles giving an initial warning. IRL you arent gonna go "more defensive" because you have a launch warning than without. You are going to assume he has launched unless you can absolutely prove he hasnt, initial launch warning or not.
In the DCS environment, against lower SA opponents it would be somewhat useful, but IRL, or even in DCS against opponents with decent SA and knowledge of stuff like MAR I doubt it would be very effective.