Not a complete stranger. An official airline employee that has been sanctioned to speak on behalf of the airline.
When you phone up SWA, do you say afterwards: "I spoke to Cindy", or do you say: "I spoke to SouthWest"? Do you say: "I bought a ticket from Cindy". "Cindy has a stupid baggage policy". "Cindy killed my dog". I bet you don't. Instead, you bought the ticket from SWA. SWA has a stupid baggage policy, and SWA killed your dog.
Similarly when someone stands up there wearing a uniform and judging whether 2 legally married people constitutes a family or not, it's not Cindy doing this, it's South West.
And if this is not reflective of South West's policies or values, they do need to come out and say it.
I'm not saying that this happened at all. Maybe the agent never said anything that resembles or truly implies: "You are not a family". But if she did, and she was my employee, she would be fired. And that has nothing to do with politics - if she said: "Gun toting rednecks aren't welcome", she'd be fired as well.
There are corporations who proudly display and enforce a single religious belief, and only want employees and customers who agree with that belief. That's fine, but I don't think SWA is one of those organizations.