Just a single data point for you - Louise and I have lived at two airparks - one was “private owned, private use”, and the current one is “private owned, public use”. Teh first was a short, narrow runway emmnbedded in the middle of a suburb, and we controlled users (invite only, no touch and goes) to keep peace with the surrounding neighbors. Private/Private seemed appropriate, but we ran the risk of neighbors and/or utility companies erecting trees or obstacles in our approach path.
Our current runway is huge by comparison - a mile long and 75’ wide - and we’re surrounded by desert. We have no issues at all with public use, and we have the occasional flight school user from nearby Carson City, as well s low approaches by Navy rotorheads from Fallon. We are protected from encroachment by obstacles erected by power or cell companies, and our county likes to feel like they have a resource that can be used for emergencies. We have one or two residents who occasionally express the though that they “live on a private airport, so why are others allowed to use it”.....but it works just fine, we are lucky if we average ten operations a day, and we enjoy the occasional landing out by sailplanes, or visits by friends who don’t have to call ahead for permission.
I don’t know how much we pay for liability insurance, but it doesn’t seem to be a significant factor.
In short, we’re happy to be Private/Public - it works fine for us.