As soon as the white landlord posted the whites only sign, he violated every FEDERAL fair housing discrimination law in the nation.
If the landlord wants to charge rent to his tenants, he gives up all his claim that the pool is private property.
The landlord can have one or the other but not both.
And why the state of Ohio should pass preliminary judgment may be a matter of States right formality, but the very second Ohio rules for the landlord, the Federal justice department will simply take over jurisdisction, and dope slap the State of Ohio and the landlord at the same time. There are fines and penalities that can result fron violating civil rights laws, and the whites only sign makes it a slam dunk case..
Now on the other hand, a totally private party can build such a pool and invite guests over free gratis. And post whites only signs with inpunity. But as soon as such a party charges a single nickel of fees, it becomes a public accomidation subject to all Federal civil rights laws.