One thing I do wonder about is the start of the third segment and the priority it should be given
Same priority as the other bits of the procedure ?
Often the third segment is limited by the engine limit .. ie push the acceleration higher and you push out the 5 minutes. In the very occasional event that the failure is real, it probably remains academic.
I can't ever recall having looked specifically at arbitrarily increasing the accel phase for a straight ahead flight path .. although I would expect that it would not routinely present an obstacle problem.
HOWEVER, if there are any TURNS in the procedure, then the crew had best follow the procedure fairly accurately as the turn radius - hence trapezoid coverage - depends critically on speed - which is linked to the configuration. A planned second segment turn at V2-ish sees you over totally difference terrain if you are accelerating to final climb speed during the turn.
Apart from the IAs for a major problem (fire and severe damage types of problems), there is probably no real urgency to do the systems drills in preference to flying the aircraft .. why not just fly the procedure and then, when the procedure workload permits, worry about the switches and PAs, etc. It seems to me that the main risk in this situation is running into some cumulo granitus so that is where the risk mitigation efforts ought to be directed ?
Unless things have changed in recent times, if I think back to 727/737 it was never a problem to fit the drills into the flying .. indeed, periodically in the sim we would do the OEI departure exercise single pilot just to put a tad more load on ourselves for practice... the workload was never a problem (unless one's mate in the back seat started playing the organ to relieve his boredom ?)