> the atmosphere creates too much drag to get there in one push

This is not true. Feasibly designs have been worked out. The challenge is the trade off between cheap, short and high-G systems and expensive, long and low-G systems.

The second push has nothing to do with atmosphere. It's because orbits repeat. As soon as the push is done, you are in the initial orbit heading away from the surface. In other words, your initial orbit intersects the ground and it would be unwise to complete a full orbit.

The second push converts the narrow ellipse of the initial orbit into a wider orbit which avoids the ground-penetrating aspect of the first orbit.

The notion of sending volatile fuel sufficient to make the correction and yet somehow not kablooeying at launch seems like a hard problem to me.

The notion of using a catapult instead of a rocket for the first stage seems safer.