Why do planes not fly "backwards" (up from the planes POV) when in a vertical climb?
Short answer: Yes. But only for a moment.

As you suggest, let's focus on the 90° case, because many things will become simpler. Intermediate climb angles can be covered by a linear superposition of horizontal and vertical flight.

Now we shoot up at 90° flight path angle (engine thrust permitting) and leave some angle of attack for the wing to create lift. Lift acts orthogonally to the local flow direction, so the lift will pull the aircraft sideways (up in the aircraft's frame of reference). Since there is no other force which will oppose this lift (drag, weight and thrust all act at 90° to the lift, parallel to the flight path), the aircraft will accelerate sideways (again, up in the aircraft's frame of reference). The angle of attack will immediately decrease and lift creation will stop &mdash; or the pilot forces the angle of attack to stay constant by pulling the elevator, and now the aircraft will further increase its flight path angle.

That is a complicated way of saying that the pilot forces the airplane into a loop. If he wants to maintain this 90° flight path angle, he better avoid creating any lift.