IANAL either, but I would think they
could ban carry during their event under the color of the language "...or agent thereof...." When they sign whatever to use the facilities, they would seem (to me) to assume some sort of proprietary control over the property for the purpose of their event. In any case, TSSAA events are also school events, which are also no-carry under TCA.
I live in Murfreesboro. I am willing to have TSSAA post their Spring Fling events "no-carry" for a few days under whatever works for them to have the City Council allow me to carry in local parks the other 360 days of the year. Perhaps that is not the "purist" view, but I'll take it.