Saying, belief in ghosts implies belief in the supernatural, and god is supernatural, therefore it makes no sense to deny god, doesn't follow for me. Belief in ghosts implies nothing but a belief that something exists which we call ghosts.
I believe in ghosts. I've seen them. I don't know what they are. Calling them "souls of the departed" is handy, and may even be true, but it's no more correct than calling them "residual bits of energy". Which also may be true, or not. I don't know enough to say what they are. I also don't know what god is. I don't see why I can't believe in ghosts without believing in god.
I believe in a Divine Presence in the universe, but when people ask if I believe in God, I usually say no. Because I don't want to be mistaken for a believer in any particular God or a follower of any particular religion. I don't want that, simply because I'm not one, and I dislike being misunderstood. I believe that the universe is god, and everything in it is god, and god is everywhere, everywhen, and everywhat. I believe god is the Divine Consciousness that is this universe, much as I am the consciousness that is this person. (Obviously, not an exact parallel. And obviously, god's consciousness is nothing like mine.) After I accepted that I wasn't ever going to be a Christian, I looked for many years for a religion that fit me, or at least a handy label to define myself to others. The best I found is "pantheist". And that's hardly helpful, because most people have never heard of it!