Take an imaginary gradient from the end of the runway rising at 6% along the path of the SID. If the departure requires a minimum % gradient (for example 6%) it means the aircraft should always be above that notional 6% slope.
At initial rotation the aircraft will be climbing at significantly more than 6%, with a high VS and low ground speed which will give a significant angle of climb, it will therefore be well above the slope it is required to be above by the time you reach acceleration altitude. You then speed up, clean up and carry on.
This all assumes all engines operating. In the event of a loss of the critical engine, then the requirement to achieve the 6% is no longer there. Often the 6% is for noise or airspace reasons, and not always for obstacle reasons. If however, there is an obstacle within the climb out, and if single engine the aircraft can not clear it, that is when you will have an emergency turn procedure on the departure. Otherwise, you follow the company standard engine fail procedures.
Generally SIDs are written to be comfortably achievable with commercial aircraft with no major concern, and if there is a challenging SID then usually additional notes, either on the plates or in company ops manuals will draw attention to it.
Hope that helps