I'm not claiming they should know the intracate details of the website. I am, however, asserting that they should be able to tell me "Yes, there's a known issue" or "No, we haven't heard that before" and assure me (even if they're lying) that they'll open some sort of trouble ticket.

This is the phone number I was told to call, on southwest.com, when southwest.com produced an error message. I don't think it's at all unreasonable to expect the person on the other end of that phone to be able to tell me if they know of any existing bug on southwest.com.