What would be the first causal factor responsible for me falling off a platform at a high enough velocity?

Yes, it would likely be the wind that causes you to fall off.

If there is no atmosphere, you would stay on the platform indefinitely, if the acceleration is indeed sufficiently low. This situation is no different from a uniformly accelerating spaceship, except what is accelerating you is not the force exerted on you by the seat you are sitting in but the friction between you and the platform.

I suppose if the platform is moving in one direction, it would eventually get far enough away from the planet that the gravitational acceleration would be too small (or even in the "wrong" (i.e. not downward) direction) to provide the necessary friction, but maybe that's outside the scope of what you were asking for.

If the platform is not moving in the same direction but always moving "horizontally" so as to maintain the same altitude, increasing its horizontal speed in the process, then eventually the platform would orbit the planet so fast that your weight would not be sufficient to provide the centripetal acceleration you need to stay on the platform, so you would eventually fly off. But to move in this way, in addition to its tangential (or horizontal) acceleration, the platform would need to accelerate toward the center of the planet by a non-negligible amount: the point you fly off is when this acceleration reaches $g$.