Probably, yes. But at least private entities can be (and frequently are) killed by customers starving them of their income - this can happen passively by simply not buying their products anymore. Public sectors ones need politicians to actively invest political capital in not just removing it, but figuring out what it was meant to do in the first place, how to do it better and then setup a new entity that's better - so there's very high friction to take on the task. Obviously, also, anyone who takes on TSA opens himself up to being labeled "soft on national security".
Of course, there are many anemic private companies that are kept alive for various reasons long beyond their reasonable lifetime, but generally this holds. Private companies come and go much faster than public sector ones.
That said, I'm not sure airport security is necessarily a good fit for a private company.
I think a good example to point to would be Pinkerton security, which for example guards some military bases when needed for things like air shows. They are highly trained at security. The TSA seems to hire almost anyone that is breathing and hasn't been to prison. They seem completely unconcerned with security and more concerned with being rule-bent weirdos.
I'm not saying that the guards couldn't be privately employed (which is the case in SFO, for an example) - as long as they have to enforce TSA stupid rules, that doesn't matter (anecdotal evidence suggests the guards in SFO are friendlier than average, but this can have many causes).
I was referring to making security a private concern in the same way operating an airline is a private concern - that I struggle to see how would work: How do you align the long-term interest that a terror attack doesn't occur with the short-term horizon of an industry that bankrupts itself every decade?
On the other hand, FAA seems to be mostly sane and evidence-driven, so they might take on the task of creating and maintaining the regulatory framework that private airport security would operate under.
