Sometimes you just cut to the chase: back in the day was trained to use cross controls TO THE EXTENT REQUIRED to keep the wings level/rolling down the centreline and maintain that cross control on rotation until well clear of the ground when aerodynamics - as if by a miracle - will gradually ease off/eliminate the cross controls.
Been lucky using same for 29 years - if it ain't broke don't fix it IMHO.
Only times I've been truely caught out and immediately "scared" was training new Captains when they seemed to have the right idea on a croswind take off only to either not be aggressive enough in maintaining the wings level OR, worse yet, backing off the cross controls when initiating rotation and wondering why the downwind wing headed south bigtime.
Good thing was tended to only happen once as they scared the sh*t out of themselves as well!
Cheers.