I would put a tenner on the two methods (constant zero-wind heading, and flying the GC track) being exactly the same
I thought the same for a while but having done the trig I'm now with the sailors!
Constant zero-wind heading gets you there in the zero-wind time (the wind component just cancels). Flying the GC track takes a factor of 1/cos(drift) longer (that's the factor by which your groundspeed is reduced). That means of course that the result is second order in the drift (i.e. if the drift is 0.1 or 6 degrees, the difference is only 1%).
In the real atmosphere of course, nothing is predictable enough to make this a sensible strategy. But a fun question, Fuji, thanks.
