Can't see the problem with airlines doing it, simple adjustment to T&C, "no jab no fly" and documents checked at check in/boarding, exactly how its done to insure revenue for the flight. Have no proof, don't fly, better have good excuse or lose ticket, no refund, this is already done for other medical conditions that are prohibited for flight without a doctors cert. Cost to airline, Nada, as its already done for revenue assurance. In fact there's actual cost benefit to the airline as preventing spread of communicable diseases will save the company in sick leave costs overall.
Cost of setting up TSA style security at state/fed cost, $5-$20 per passenger head tax or from public tax revenue of similar amount.
What would I prefer? Pretty sure the answers obviously the free of cost option.