@Robert Lenfestey I wouldn't be worried about someone saying you're lying. Simply not saying something doesn't make you a liar. And even if you were to negate the fact that you own a place when directly asked: so what? It's really none of other people's business. They have a contractual relationship with an LLC (a legal entity) that manages a property (assuming you're doing it right). That's what they need be concerned with, nothing else.
People should really understand the difference between doing something in your own name (as in: you as a human being) and then doing the same thing in the name of an entity: every entity needs some actual human being acting for it. Still, that human is doing it for the entity, not for him-/herself! And this would not change if you are also the sole member of said LLC. It's a legal principle (often called "corporate veil") and the whole point of setting up an entity in the first place.
Regarding your goal of not publicly displaying your name and private phone number etc: one of the most legitimate things to do in this business (= "landlording"), as far as I'm concerned! So by all means, yes, do this. You will see many, many other people recommending to do just that. Just be consequent about it as also @Account Closed correctly stated.
Finally, which kind of entity you chose is, as others have already mentioned here, typically primarily tax driven. There is no such thing - from a legal point - as an S-Corp. That's a mere tax related election (if you fancy it, you can start reading Chapter 1 of the IRC - I wouldn't, LOL!). The entity in this kind of business will, most likely, be an LLC either way. But from a taxation point of view there exist different types of LLC. Even a single-member LLC with no special election ("look through for tax purposes") will still remain - from a legal point of view - an legal entity in the form of an LLC.
I don't know if this is confusing or not, but in a nutshell: one should always distinguish between the legal aspects and tax related aspects, especially when it comes to (legal) entities.
Edit: ups, just realized that this thread is over a year old... oh well.