That is turning out not to be true in the part of the market that matters.
Bag fees aren't charged to an airlines most important customers (i.e. golds, pax with paid, business-men with just a roll-a-board, etc.) so all those cute SWA commercials and the signs on the planes mean nothing to them who are still the bread and butter of the legacies.
Ironically, charging a bag fee lets the airline waive the fee, which increases the importance of being a gold or whatever at a legacy, making it even less likely that SWA will steal them away.
As far as the people on the rest of the plane go, they are the most price sensitve and if they see a $199 at SWA and a $169 at Alaska, a large number will almost instinctively buy the ticket on Alaska even though if you travel with 2 bags you are effectively paying more on Alaska ($220).
To make matters worse for SWA, the businessman that is only traveling with his carry-on is going to make the Alaska fare a real bargain.
I don't think SWA's next step from here is a bag fee though. I think SWA's next move will be a super-discounted fare, if you agree to not fly with any bags. Kind of a reverse fee.