I don't think so, as you seem to have misunderstood my "bright idea."
It's not airlines I was talking about. They'd have nothing to do with it. Just as they have zero to do with security checkpoint screening now. Rather, it's multiple, private replacements for TSA, each of which would serve all the airlines/gates at an airport.
What's more, even before the TSA existed, the airlines didn't do the screening. It was a private security contractor hired by the airport.
It's, as you correctly imply, all about incentives.
In my "scenario" these hypothetical "competitive private replacement" security screeners are paid by the numbers of bodies it passes through.
And so I'll ask my sarcastic question again. This time specifically to you:
What could possibly go wrong?
Edit: Clarified prose.
Very quickly “no security airlines” would go out of business as no one would want to risk their life needlessly, especially if the lack of security led to an incident.
However an airline that used nonintrusive, convenient methods for security, fmight find a loyal passenger following.