Probably. Maybe not. They tried to deal with Disneyworld like the rest of the state...and, I'm not sure that's the right thing to do.
Floridians are aware of this, or learn quickly...frankly, most Southerners would be aware. But, this family was from Nebraska. Or, just imagine some British family?
http://www.mirror.co.uk/news/world-news/terrified-brit-family-chased-alligator-8197003
I suspect, however, that the beaches will be severely altered and or shuttered after this...and they'll call that that...but, it doesn't stop issues in areas like, say, Ft. Wilderness, etc...
Frankly, imho, with a property like Disney, they should be allowed to remove gators upon discovery, at least in certain waterways...and, the State flat out admits/implies that is not the case. As their regulations for the rest of the state, the gator needs to be deemed a "nuisance" first, then the contract approved, and that could take days if not weeks...