If gravity just emerges from the curvature of spacetime, is it actually a force? Why is it one of the 4 fundamental forces of nature?
It is certainly true that our (classical) theory of gravity describes gravity in a geometrical way as the curvature of spacetime, and this seems very different to the other three forces. What is less well known is that the other forces can also be written as a geometrical theory and described using curvature. This formulation is known as a Yang-Mills gauge theory.
Indeed, Maxwell's theory of electromagnetism is actually a classical Yang-Mills theory, though Maxwell didn't realise this as Yang and Mills only developed their ideas in the 1950s.
Things are more involved than I've suggested, because the electroweak and strong forces are quantum field theories and don't have any classical analogue like classical electrodynamics or gravity. For gravity, of course, we don't have a way to quantise it. I should also point out that gravity cannot be written as a Yang-Mills theory, so the concepts of curvature used are different.
Nevertheless, your claim that gravity is different to the other three forces because it's based on spacetime curvature is not as obviously true as it appears at first glance.
It could be or it could not be. It's really not clear. You still need some sort of means to communicate with space-time and tell it to curve. This is where gravitons could potentially come into play as the force mediator, and that would technically make gravity a fundamental force. It is certainly expected to be quite different from the other 3 forces. But I'm certainly not an expert in GR and could be mistaken.