Well, just to clarify, if your engine is idling, the injectors are opening. They're just running a vastly smaller duty per cycle. So they're open for a shorter period of time. If the injectors remain closed, your motor isn't running.
As for the mission; there's still fluid flow. The pressure is mostly maintained by the ECU. The issue is the sudden gear engagement when you need to accelerate again.
It's like trying to clutch start a manual: dumping the clutch in gear while rolling is bad news. Same issue with an automatic. Engaging those "gears" (they're clutch packs, really) while moving is sudden force on those parts.
I advocate leaving the car in gear, using a combination of engine braking and the cars actual brakes. That should help minimize wear on either part.
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