So if I understand this correctly, the key has to slide in the keyway?
The key should fit snugly in the shaft. This is why a properly machined shaft comes with the key already snugly fit. It should slide in the prop but not be sloppy. The idea is that when the prop is sliding onto the taper the key does not move and stop the prop from seating fully on the taper. This is why you always install the prop first, without the key, and mark where it sits on the shaft with a Sharpie marker, then remove it and re-install with the key. If it does not hit the Sharpie mark then it has become key bound and the fit is not correct. All of this occurs after properly lap fitting the prop to the taper.
Can it also be too loose? I always thought that the key was there to transfer all the torque and that the taper kept the prop aligned.
Yes a key can be too loose and if it is, and the prop not properly fit, it exacerbates the
shock loading of the keyway when going into gear. With a properly fit prop the taper is what is taking the load, the key is added insurance. I once removed a prop that had been installed for approx 7 years and it had no key at all. What it did have was over 85% contact and a good taper fit.. Someone probably went to lunch and when they came back never removed the prop and re-installed the key..

The prop had not budged and was still extremely difficult to remove even with a beast of a puller.
Here is a tapered coupling end cut-away. Picture this as a prop, as opposed to a coupling, and allowing the key to slide UP the shaft while fitting the prop. If it does the taper will NOT mate as it should and the key will be taking the vast majority of the load, which it is not designed to do.
This is another failure due to improper fit. The diver was able to recover the 3K prop at a cost of about 1.2k.....
This is
spooning of the keyway..