I believe the weapon has to be unloaded and in a enclosed carrier thus requiring two steps to fire it, removing it from the case and loading it.
Nope. Florida law is quoted above - nothing says it has to be unloaded, only enclosed, and "enclosed" is defined rather gently, to include the glove box whether locked or not.
So the answer to the OP question: Can I keep a loaded handgun in the glove box of my truck without a CWP in Florida? Clearly and expressly, yes.