I wasn't under the impression that the government making things that help (in the case of the TSA, that last word is highly sarcastic) a private industry render that industry public. If that were the case, any industry that relied on the interstate highways (or even just public roads) would seem to apply. Or anything involving oceanic operations (Coast Guard, NOAA weather stations, survey charts, hell even GPS). In many parts of the country, municipal water is government-run; does that mean that the government "funds" restaurants by making sure that they don't need to perform their own water quality testing before they can serve it to their guests?
Mind you, I'm ignorant of (and therefore not considering) any legal history which may exist around this issue. So far as I know, the gov does not actually directly fund airlines in any way and instead just has agencies who are responsible for making sure that planes don't fall out of the sky, crash into each other, or get hijacked for use as missiles... at least, not very often. I'm not sure how this is different from the government agencies which make sure that road signs are accurate, bridges can support the weight of a truck driving over them, and people aren't allowed to tear through residential neighborhoods at whatever speed they feel like. Well, aside from the fact that many (though not all) of the relevant agencies are operated by lesser governments than the feds.
Anyhow, if there's actually something that makes the difference, then I'm curious but accept that my understanding was incorrect. To the best of my knowledge, an airline agent is a public servant in much the same way that a cruise ship agent is - that is, not at all. They have authority within the scope of their employer (including the authority to evict you from the plane, assuming that in doing so they are not putting anybody's life in undue danger) but then, so does any restaurant proprietor if they have reason to believe that you are harmful to their business. They have the authority to call the police, but so does any private citizen for any scenario in which they think a crime has been or is being (or imminently will be) committed. They do not, to my understanding, have the authority to force the police to arrest anybody or indeed to show up at all, although given the nature of their job I'm sure they can convince the police to show up any time they want them to (at risk of crying wolf too many times). Again, correct me if I'm wrong about any of this.