Flyjet:
You're right. P-Factor does not exist when the aircraft is at a precise zero degree AoA. It receives far too much blame for yaw tendencies during the groundroll portion of the takeoff phase. The real culprits behind yaw in this stage for a nosewheel-equipped airplane are the corkscrew-shaped propwash striking the empennage, torque produced by the engine itself (Newton's Third Law), and, as has been mentioned, the crosswind direction. P-Factor and gyroscopic precession are of primary concern for taildraggers in this stage.
The short answer to your question is that -
on the ground - there is no "critical engine" as it relates to P-Factor. If the engine fails in a twin and you're on the ground, you have two options: Continue the takeoff roll until airborne (at which point P-Factor and the critical engine become
major issues), or chop the power and stay on the ground. The wisdom of each course of action is beyond the scope of this discussion.
This website did a great deal to broaden my understanding of yaw during my CFI days:
Yaw-Wise Torque Budget [Ch. 8 of See How It Flies]
EDIT: It's also worth noting that most nosewheel airplanes are designed such that the propellers are not at a precise zero degree AoA while the airplane is on the ground, so P-Factor still exists. It is negligible in this stage, however, compared to the other factors listed.
I hope that, between my disjointed explanation and the nigh-impenetrable wall of text on the other side of the link, you've got a satisfactory answer.