Motor overload protection is a generally good idea anyway, and current sensing is usually a good way to do that. So if a finger in the way of the door would cause a measurable increase in motor current before serious injury then that could be a reasonable way to go. This depends on the load and mechanics, though, if it's a very light mechanism then maybe it would never cause injury anyway, or if those doors weigh a couple of tons then the difference a finger would make is well down in the noise. So you'd have to be somewhere in the middle. It's possible to work around the startup current by either doing a soft start (assuming the load allows this), or disabling the overload protection for the initial startup, or only tripping the overload when the current is over the limit for a certain period of time. Having some compliance in the mechanism (whether deliberately added in the form of a spring, or just inherent flex in the mechanism) may help. If you have position feedback, you can schedule the current limit as a function of position to account or any changes in actuator load with position (like if the mechanical advantage changes as the load moves), which allows a tighter current limit.
One extra thing to watch out for with overload detection is the dynamic behavior of the load. I have a motorized desk that has overload protection and sometimes when it starts moving--especially if I haven't moved it in a bit--the jerk of the initial stiction breaking loose will make my monitor arm bounce up and down enough to trip the overload.