You might as well ask why an airplane doesn't just start sliding sideways, rather than flying in a horizontal circle, when it banks (while increasing lift as needed so that the flight path remains in the same horizontal plane.) The two situations have a lot in common. The key attribute of a circling flight path, be it a horizontal or a vertical circle, is that there is constantly a net centripetal force acting perpendicular to the instantaneous direction of the flight path. This net centripetal force makes the flight path curve, or change direction. Meanwhile the aircraft has an inherent tendency to stay generally aligned with the instantaneous direction of the flight path, albeit with some offset angle. (Oversimplifying a bit, we call the offset angle the "sideslip angle" in the case of a horizontal turn, and it tends to stay near zero in most aircraft, while in the case of a vertical turn we call the offset angle the "angle of attack".)
By the way, keep in mind that in a linear climb, as opposed to during the initial pull-up or during the start of a loop, lift is less than aircraft weight, not more. For more see related ASE answer Does lift equal weight in a climb?.
Keep in mind too that though you are right to point out that the lift vector is defined to act perpendicular to the flight path, not the longitudinal axis of the aircraft, it would be possible to re-define the lift vector so that it was tied to the "body frame" of the aircraft, but that wouldn't fundamentally change our view of what is happening during a loop.
You may find it helpful to first think through the problem with the simplifying assumption that the aircraft's rotational moment of inertia about the pitch axis is zero. This basically means that the aircraft is going to instantly assume whatever angle-of-attack the pilot is "setting" with the control stick, which is fundamentally an angle-of-attack controller. You can seen that it is impossible for the flight path to stay linear if the net force is not zero, i.e. if there is a net centripetal force on the aircraft. Once you are thoroughly comfortable with this concept, then you may wish to also consider how the aircraft's actual non-zero rotational moment of inertia about the pitch axis might slightly complicate the stick inputs required of the pilot in various maneuvers.