There's one key item that's missing from any/all of these suggestions: security liability. Below is merely presumption on my part so grain of salt and all that.

Before 9/11 the airports and airlines that operated out of them were co-responsible for security (the airport hired the screeners and charged the airlines the gate/terminal fees). When "everything" hit the fan that day, suddenly you had everyone (airport operators to airlines themselves) scrambling to minimize/mitigate the lawsuit risks from such a huge security lapse.

What's forgotten is there was a significant change in liability assignment as part of the Patriot Act that established the DHS/TSA and thus the government assuming the liability for all commercial airline security. I even want to assert there was a retroactive immunity applied to the airlines solely based upon the potential of deleterious effects on their ongoing operations should negligence lawsuits against them proceed.

So yes, "eliminating the TSA" would not happen without the reassignment of liability back to something else other than the government and somehow I don't think you'll get too many commercial airlines to agree to that.

To note, some have suggested that any NGO would have an issue with the legality of the administrative searches (as they've been dubbed) but as before 9/11 this had already been hashed out in established case law decisions.

With that in mind, this establishes a perimeter you would need to operate within while maintaining liability with the government.

Transitioning the TSA away from the intelligence/law enforcement branch of the cabinet would go a LONG way towards alleviating a lot of this scope creep from a search for WEI to all manners of criminality.

I'm reminded of how the FCC has been given the ability to establish rules & policies regarding "communications" within the US however they're stymied at numerous points in their charter by the multitude of entities that are associated with "communication" - Internet service providers, ad-supported broadcasters, pay-supported broadcasters, end users. All parties with a vested interest in the outcome are (generally) involved in negotiations and suits are filed, settled, or decided to create the path Congress (perhaps not 100%) envisioned.

Why can't we have the same thing with airport/airline security?

Somewhat tongue in cheek: the customer is locked into a duopoly of choosing between "Charter Airlines" or "Verizon Airlines" depending upon what level of security they've chosen to apply to their flights.