Why is the force that we feel the normal force, and not the net force?
This is part of the key insight that led to the development of general relativity (GR). Einstein called it his “happiest thought”. Today it is known as the equivalence principle.
For a little background, we are all introduced to physics with Newton’s laws. What is covered only much later is that they only apply in inertial reference frames. In accelerated, rotating, or otherwise non-inertial frames, we have to introduce so-called fictitious forces. Fictitious forces cannot be directly measured or detected by any means. They can only be inferred by their effects in motion in the non-inertial frame, and they can be made to go away simply by transforming to an inertial frame.
It turns out that gravity is a special case. Uniform gravity is a fictitious force, only tidal gravity is a real force. Uniform gravity, like any fictitious force, is completely unmeasurable. Any time you think you are measuring uniform gravity, you are actually measuring something else, usually the normal force as you discovered. So in problems like this, where the distances are too small for tidal gravity, gravity is a fictitious force in the sense that it has no measurable effect. It causes no measurable anything, it only is used to explain the motion of objects.
For example, an object on a scale registers a 100 N force. So there is a measurable 100 N real force pointing upward. If we were in an inertial frame, like a space station in deep space, then they would be accelerating as they go by us, like if the scale were on the floor of a passing rocket. But if they are observed not to be accelerating, like with us on the ground observing the scale also on the ground, then we infer the presence of a fictitious force pointing down. This fictitious force has no directly measurable effect and is only inferred from the fact that the object is not accelerating.
This is the equivalence principle. Only tidal gravity is real (it is spacetime curvature). Locally gravity is a fictitious force that has no measurable effects at distances too small for tidal gravity to matter