(The implication being that if you diverted you must finish your flight to the original destination to meet some requirement external to the Expert server interactions between the pilot and the controller.)
And the answer is no. If you divert, you divert. That’s where you land. If you take off again from JFK that is not the slightest bit different from any other flight taking off from there and you will be held to the same standards of the TFR as any other plane.
There is not a loophole to the TFR. Nothing outside of normal communication between pilot and IFATC is afforded any consideration, whether that is completing some sort of mission for a VA or something a third-party app does.
Believe me, anyone reported for doing such a flight will take note of any and every pilot they see allowed to make the same flight.
If it isn’t enforced across-the-board, it can’t exist, because we can’t tell pilot X it’s okay for pilot Y but not you because he has to do this or that or another app simulated an engine failure.
The only thing that matters is “are you increasing the traffic volume in a conjested area with a unnecessarily short flight.”
The reasons behind that flight are irrelevant. It’s either yes or no.