One reason (in the UK anyway) why we give "Climb straight ahead" is for the Tower controller to separate the departure from other traffic (possibly Special VFR) under their control - and as an Aerodrome controller he/she cannot use any type of radar equipment, and therefore cannot give headings. So to guarantee separation by using this method I expect the pilot to take the drift into account and track the extended centreline.
If, on the other hand, an "after departure" heading is given to the Tower controller by Approach Radar in order to separate from Approach's traffic, the radar controller should take the drift into account when requiring the aircraft to track straight ahead.