(1) A single agency can have within its mandate, expertise, and budget the security apparatus, which requires full-stack authority: airport design and administration, customs officials, airlines, planes, luggage, etc. The airlines can focus on the task they actually want to perform and are qualified to perform, whereas the one agency responsible for security can be audited and held accountable for that task as a unit.
(2) Customers cannot purchase less security. This is an advantage. It is nonsense for a customer to walk openly onto an aircraft carrying a rifle.
I think "airport security" has much less to do with passenger security than the risks inherent with allowing planes to fly through our skies. 9/11 is a clear example that many more people than the passengers of the flight can be affected via air-terrorism. This is why planes are controlled.
I'm not opposed to that control; I just marvel at how awful a job the U.S. government does at determing the cost/convenience/effectiveness tradeoffs. In my experience in the United States, this seems to be a systemic problem with governmentally-administered services that does not seem to exist so much in other governments.
There aren't many airlines, and I expect the overwhelming majority of consumers would consistently purchase the cheapest option. If airlines are the agency entrusted with safe flight, this makes everyone less safe. Rather, airport and flight security seems better serviced as an airport-surcharge or tax. What's missing is effective, accountable security provided by experts that we trust. I don't know why that isn't something we can solve.