What about the (county owned) fairgrounds within a school zone?
If your goal is to prove that the law doesn't say that it's illegal to possess a firearm (with certain exceptions) inside of a school zone, then you are doomed to failure. The law is clear and other than the listed exceptions, all of which have been discussed, does clearly prohibit possession.
On the other hand, if you are truly interested in the minutia of how county fairs are operated and how that might relate to this law, I've already provided a hint on what to look at with my earlier comment about how an organization can rent/lease public property and essentially transform it into private property. I've done just a very minimal study of the topic myself and found that at least some county fairs are actually operated by non-profit organizations that lease the property from the county to hold the fair. That would effectively make the property private even though it is owned by the county.
If you really want to get an authoritative answer to your question, here is the first step. You contact the specific county that owns the property you are curious about and ask them how the fair is run.
Does the county itself run the fair or does a corporation/organization lease the land and run the fair?
If a corporation/organization leases the land and runs the fair, you find out what organization does it and if they are a governmental organization or not.
If the county actually runs the fair themselves, you find out if they have leases with the individual operators who run specific rides, food booths and exhibits or do they operate all the rides and exhibits themselves?
Until you get that information, it's really not possible to do anything other than speculate.
Once you have that information, you can provide it here and see if someone with a legal background is willing to take the information and dig into the law to see how it applies to the specific situation. Or you could take it to a lawyer who specializes in such things and pay for an opinion.