Why would a spinning space station create a centrifugal force on an astronaut rather than simply spinning around him/her?

Put a stationary astronaut in a small room inside a large spinning cylinder. After an instant walls of that room will hit him, and suddenly he will have the same velocity as the room. Due to angular motion, the room accelerates towards the axis of the cylinder. Subsequently, through the support force from the floor (the floor is at the surface of the cylinder) accelerates the astronaut too towards the center of the cylinder.

If the room accelerates $9.81~\rm{ms^{-2}}$ towards center, this will be feel like the regular gravity.

Note that one cannot feel gravity or acceleration as such (except for tidal forces). The 'weight' one feels is the support force from surfaces. In other words, gravity feels like so that you are constantly being pushed by the floor, which accelerates you at the rate of $9.81~\rm{ms^{-2}}$. If you stand, your organs will be pushed down etc.

You are correct in that if the astronaut is undergoing no translational or rotational motion relative to the centre of rotation of the space station the astronaut will feel weightless as in diagram $A$ and will not touch the space station.
This is equivalent to jumping onto a rotating turntable with no friction acting.
That feeling of weightlessness is due to the fact that the astronaut feels no contact forces acting as the astronaut is not touching the space station.
If the space station is rotating what the astronaut will see the space station going past so the astronaut will know that the space station is rotating but the astronaut will not feel any contact forces due to the space station.

If the astronaut comes in contact with the space station as in diagram $B$ then a frictional force between the feet of the astronaut and the space station or the astronaut holding on to the space station will cause the astronaut to rotate with the space station and so the space station will exert a centripetal force on the astronaut to accelerate the astronaut.
So the astronaut is now subjected to a contact force either at the astronaut's feet or arms which is the same as as if the astronaut was on the Earth if the space station's rotational speed is correctly set up.

For example a space station of radius $160$ m would have to rotate at approximately 2.4 revolutions per minute to simulate a gravitation field strength of $10$ ms$^{-2}$.

To go back to a permanent weightless condition (no contact forces) the astronaut would have to engineer it so that the astronaut is undergoing no translational or rotational motion relative to the centre of rotation of the space station.

If there is no atmosphere, and the station is a relatively smooth cylinder, you can indeed float there as the exterior walls spin around you (in the middle, or just above a wall, or anywhere).

Now, suppose you start drifting towards a wall (maybe you threw your shoe the other way). You move towards the wall, but do not accelerate due to the rotation of the station. However, the exterior wall of the station is moving very fast as you approach it. You hit the wall, and both bounce and gain some horizontal speed in the direction of the wall. You'll also start to spin (as it will mostly produce torque on you).

This new velocity vector results in you intersecting with the wall again somewhere else, with more velocity normal to the wall and less horizontal usually. Each time you do so, you'll gain more and more angular momentum -- you'll spin both around your own center of mass (lacking a way to somehow counter it), and in a sense around the center of the habitat.

Assuming you avoid getting mashed to a pulp in your repeated hitting of a high velocity wall, you'll start travelling along with the wall more and more. As you do so, the wall will start seeming "more down" and less "moving fast", as your impacts with the wall will be closer to normal with the wall, and less glancing blows with something moving very fast.

If there was air in the station, the air would be moving along with the wall (for similar reasons as you'll end up doing so), so it will drag you along, like a strong wind. This drag will result in you falling faster towards the wall as you match its rotational velocity, compared to the above scenario -- basically, it moves the repeated collisions up in time and space (up, as in earlier, and up, as in further from the wall).

If there are large features, like buildings, those buildings will smack you on your side and speed you up to the rotational speed of the exterior wall. You can consider this a more aggressive form of wind.

Once up to speed, you start experiencing the pseudo-gravity of the rotating station when you are supported by the floor. When you are not, you'll experience pseudo-free fall, where the longer you are in the air the faster the ground will move relative to you (up to a point).

But what does it feel like, other than the being beaten to pulp "getting up to speed"?

While falling (with no air in the way) it feels like free fall. Same when jumping in the air. When "stationary" against the edge of the station, it will feel mostly like standing on the Earth.

You cannot, in general, directly feel gravity. Free fall, which you can experience to a lesser extent on a roller coaster, is the feeling you get when you are not being supported by some surface.

"Normally', you are supported by a substance, which pushes against you (the ground under your feet, the water in a pool, or the air as you reach terminal velocity). The parts supported by this substance then push against your organs and the rest of your body.

In free fall, no such support exists: you are still experiencing gravity, but not the "support".

The exception is tides, which if strong enough can produce forces directly on your body. No human has experienced tides (due to gravity) that strong.

A rotating station creates a pseudo-tide effect, because things closer to the axis have "less pseudo-gravity". If you are tall relative to the station radius, your head will feel less gravity than your feet. This could be felt directly while in pseudo-free-fall, but would more likely be felt as a tendency to rotate in free fall, or when standing on the ground.