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No, it won't, because neither the thrust vector nor the gravity causes a moment in this situation. The key insight to the pendulum fallacy is actually that this situation doesn't create orientation-dependent momentum. That would make it a pendulum, but it isn't like that, hence the "fallacy" part.

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I see what you're saying. That would imply in the absence of an atmosphere a gravity turn isn't really possible. You'd pick up horizontal velocity, relatively slowly, but assuming you didn't exert some kind of control your rocket would always stay at the attitude it had at the end of the pitchover.

Yes, what is called "gravity turn" turns the velocity vector, but if the rocket didn't actively reorient itself (so as to keep in line with the velocity vector), gravity itself wouldn't rotate the rocket (change its attitude), because almost-homogeneous gravity field won't impart (meaningful) momentum. (Satellites can actually use it that way, but the time constants are substantial and not really relevant for an ascending rocket on a timescale of minutes.)

However, this is independent of the presence of the atmosphere. In real life, both scenarios need active control of attitude.

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