Since I got lot of questions regarding PropHunt, I decided to finally test it. I used to suggest to disable the mod via blacklist, because I thought it would cause issues, but it actually works better then expected.

For Hunters Team, the model selection is working properly. I couldn't find any issues.

For Props Team, the model selection simply does not work. That's actually great, because I thought it might add gamebreaking bugs, which is why I suggested disabling the mod. After all, you're supposed to disguise as props, rather then randomly switching your model and breaking the game.

More technical: The T-posed Kleiner model you see at the beginning is not your player entity. Without looking at the PropHunt source, I think the gamemode hides your player entity (and therefore your player model) and spawns it's own prop entity (the T-posed Kleiner). It's not possible to turn your actual player model into a prop, without causing errors, due to it's skeleton and animations. That's also why the Kleiner model is t-posed and not animated: Since the beginning, you're playing as a prop, not as your player entity.

Therefore, I no longer advise against using this mod in PropHunt. It should work properly. If you find any issues, feel free to report them here.