Is being in an airport actually considered private?

It's a public space, and you must show ID to gain access to the secured area. Additionally, you are subjected to baggage and carry-on inspection, as well as body inspection and metal detection, etc. There are cameras everywhere, monitoring and recording everything.

Presumably this system was designed to recognize individuals that may be traveling under false-identities, and are known "bad guys" - otherwise the nation-state security apparatus would have known about the attempted air travel well in advance.

The ability to abuse this system may be real, but it seems much more likely your rights would be violated well before you reached any facial recognition systems.