Second: They already do float. They recover them quickly (& sometimes reuse them) if they miss the catch. But the fairing halves are 800km down range. You need to have a boat nearby anyway, so might as well try to catch it before it lands in saltwater and screws it up.
The salt water produces corrosion that fresh does not. That is why military aircraft operated near the sea get washed so much. Both inside and out, including the engines.
We would come back from overwater flights over the Atlantic and either go through a 'bird bath' which is basically a car wash for aircraft. Or they would drag a fire hose out, hook it up and spray everything down. It was pretty cool to watch the exhaust gas temps when the started dumping fresh water into the engine inlet.
Salt water is the worst. It takes a lot of effort to clean it all off and it causes corrosion all the while. Even the ones that do hit water, they get them out as fast as they safely can.
As far as I know (correct me if I'm wrong), SpaceX have only reused previously-wet fairings on their own Starlink launches. They don't risk potentially damaged parts on external customers.
Then, when it returns to land, someone can peel off the rubber sealant, do some quick quality control inspections, and authorize the fairing to be reused on its next launch.