...It would have been fun had this sort of a rescue/resurrection attempt become the TNG standard for any random injured or killed redshirt (now mustardshirt). That is, instead of just going "He's dead, Jean-Luc", Crusher or some token medic would call for a beam-out and/or slap a "medical evacuation transponder" on the victim, and we'd either see the beam-out or, if the VFX were too expensive, hear the sound when the camera moves on. And later on, dialogue would tell whether the poor mustardshirt made it or not.
Only with main characters would there be the weekly extra tension of whether the "ambulance beam" arrives in time or not, of course. Extras would typically get the beam without plot complications, but might perish off screen nevertheless.