Just a quick add-on to the original post. The claim was that UA first class might not be much more than Southwest WGA fares close to the flight time.
Just to note that when SWA opened their schedule on August 25 to include all of March and spring break, I looked into and bought flights DEN-MCO for mid-March.
SWA price: $229 each way, or $458 round trip.
UA (coach price): $757 round trip for similar flight times. Then add to that bag fees and potential change fees. I didn't specifically write down the price of first class, but quite sure it was $300 more.
So ... while the OP may have seen a case where UA-first was close to SWA-coach ... I think that is more the exception than the rule.
But, never be afraid to price a trip on multiple web sites, including SWA. You never know... but don't hold your breath.