No they don't. Southwest is not the cheap airline option, that would be Frontier/Spirit/JetBlue. Southwest is often cheaper than United/Delta/American by a bit, but it's not that much.
I share a similar experience. I’m puzzled by the common perception that Southwest offers low prices. In my case, their fares are often quite comparable to American Airlines and United. Sometimes, the flights I prefer are even more costly with Southwest. I don’t have a strong preference for any specific airline; my choice is mainly based on the quickest route with the fewest layovers and reasonable hours.
And personally I very much dislike the Southwest approach to boarding. I feel much calmer knowing that my ideal seat has been reserved.
Southwest is not a budget carrier -- they were never that cheap.
But I would pay a premium not to fly Southwest. If there was Delta flight that was a $100 more than an equivalent Southwest flight, I would take the Delta flight.
But that's the point - Delta (and others) aren't that much more. So people aren't paying a premium to avoid Southwest, because there isn't a premium to be paid.
It’s a hypothetical premium. Fares for the same route can differ between carriers depending on airport, time of travel etc. even among mainstream carriers. I would pay more not to fly Southwest.