Just to be clear, there is no such thing as a "free" bag. It is simply that the cost of carrying the bag is included in your ticket price.

If your analysis was correct, then WN should not have any carry-on issues as all tickets include two no fee bags. But, WN experiences at least as bad a mess with carry-ons as carriers which provide no included checked bags.

In addition, all three legacy US carriers provide an included bag to elites. Yet, elites are a large part of the carry-on problem.

The carry-on and accordingly boarding mess is solely caused by the failure of the carriers to adequately police the carry on rules. While it is true that if everyone showed up to board with the two largest bags permitted, there would not be enough room on the typical single-aisle mainline aircraft, the problem is vastly reduced when the problem is policed up front.

AA does a pretty decent job of this and has started pulling more and more people over with a third item or an oversize bag and having FA's onboard remove smaller carry-ons from the OH and either hand them to the passenger for stowing under the seat in front or gate checking.