Kieth,
Alligators cannot pass any more disease on than a fish could, they are both cold blooded animals. You would need to cook the foods you were feeding them to insure you were not getting any thing in that way, but that is the only difference, but it also is a cheap food source. Alligators in Louisiana are not an invasive species and every year the La. game and fish department personnel go out into the marsh and raid as many gator mounds as they can to bring the eggs into captivity to hatch and raise to a point they can be re-released. This is how they have managed to get the gator numbers back up in the southern states.
Look this is a simple discussion, in most instances to simply pass time, but it is a very viable system, probably not right for the average backyard AP person, with no experience dealing with alligators. What I will say is in the right environment, (southern states) where alligators are legal, and to someone familiar with their handling, this could be a beneficial setup. I would not be surprised if on a commercial scale you could not develop a partnership with the G&F to accept hatchlings, use them to a certain size, at which time they could come get them and return them to the wild, and provide you more hatchlings.
Concerning invasive species, I certainly would not advocate putting gators in your system in Ohio then turning them lose there where they are not native, although they would most certainly die the first winter, but I do agree with you on the invasive species problem.