Are municipal airports in the U.S. treated differently than most public places in regards to legal, holstered, loaded firearm carry?
As you hinted, it depends on the state. And the airport.

Some states allow travelers to carry their hunting rifles or pistols aboard general aviation planes. It is recommended that they buy the ammunition at their destination, if they can, for safety reasons.

Best answer I have is this:


  Any private pilot desiring to transport a firearm, weapon, or similar device must review the appropriate regulations and should contact the local airport management or law enforcement authorities for approval to land at any airport if any firearm is on board.


Source: AOPA

Another way to look at it, if you have your own private airstrip, who's to stop you? Just contact the local authorities of your destination before heading out.

For unconcealed firearms in a public airport:


  44 States allow lawful carry of firearms in the common terminal and baggage claim areas of airports.


Source: Florida Carry and CBS News (2008)

Disclaimer: contact local authorities before trying it, aka don't try this at home.