Your problem of falling elevator is probably the simplest and best example of problem with non-inertial frames. By definition, an inertial frame is a reference frame where Newton's laws of motion can be applied without caveats. For that to be true, the reference frame must be non-accelerating and non-rotating. By that definition, the reference frame attached to the elevator is a non-inertial frame - because it is accelerating towards ground. In your words, the 'perspective of the outside observer' is the actual inertial frame.

Now let's see why there is no gravity from the elevator's frame of reference. Remember that by Newton's law of gravitation, gravitational force is GMm/r^2. If you plug in the values, you won't get a zero. This means that the person inside the elevator is definitely facing gravitational 'force'. (I'm assuming classical mechanics where gravity is a force. Relativity considers it as another pseudo force. But that view won't affect our current discussion)

Consider the non-inertial frame inside the elevator first. In order for the person to be experiencing no force, all forces on him must be balanced and cancelling out. We know of only gravitational force that is acting downwards. So to cancel it out, we need to invent an equal and opposite force on the person. That force is called 'rectilinear acceleration force' and its value is given by ma (upwards), where m is the mass of the person and a is the elevator's downwards acceleration. So the net force on the person is mg-ma. Since elevator is in free fall, a=g and so the force on the person is zero.

You can see the mess here. You have to invent a new force that doesn't exist and have to be careful about its direction too. If the frame is rotating, then more pseudo forces have to be invented - the Coriolis force, Centrifugal force and Euler's force. All these 4 forces don't exist in reality, and are there purely to account for non-inertial frames.

Now let's see how the forces play out when considered from the inertial frame of the external observer. Let's consider if the lift is stationary first. Obviously the person inside the elevator is feeling gravitational force that is equal to his weight. In reality, he is not actually sensing gravity acting on him - he is instead sensing the 'reaction force'. The reaction force is the force that the floor of the elevator is applying on him to counteract his weight. In other words, the reaction force from the floor actually cancels his weight - so he remains stationary instead of accelerating towards the ground.

Now, unlike pseudo forces, reaction forces are real forces. In this case, it's provided by the electromagnetic repulsion between the molecules on the floor and in the person. This reaction forces don't just act on the person's foot. It acts throughout his body. For example, his internal organs don't fly off under gravity because the reaction force provided by the connective tissues hold them in place, counteracting gravity.

Finally what happens when the elevator is falling? The elevator is in free fall, so its acceleration is g towards ground. The person is also falling with the same acceleration g. This means that there is no reaction force acting on him to counteract the gravity. In other words, the person is no longer supported by the floor - that support was the reaction force. Remember that I said that the reaction force was the force that the person was sensing as gravity. Now that reaction force has disappeared. So the person in effect feels that he is weightless, even though gravity is still acting on him.

This is the advantage of inertial frames. All Newton's laws can be applied in terms of real and concrete forces with well known causes. You don't need to invent anything new and then justify their existence.