Everything that you say is essentially correct. When Flex / Assumed Temperature is used, the calculation considers the engine only, and provides you with a Flex / Assumed Temperature to give the equivalent thrust as though the day were actually at the Flex Temperature, and produces the reduced thrust accordingly. The calculations
DO NOT re-compute ASDR, TODR etc., thus, V1, Vr, and V2 being the same IAS, will be at a lower TAS than assumed.
From this, ACTUAL Accelerate-Stop Distance required will be less, OEI Distance to screen height will be less, and AEO margin to TODR will be greater than the nominal 15%. In short, you win in all cases. A fourth bonus is that for an OEI continued Takeoff, TAS during the 2nd segment will be less, leading to an increased climb gradient.
It's win, win, win, win all the way!
Regards,
Old Smokey