Question: What if the opposition group claims the property owner's billboard is "Hate Speech"?
Answer: Hate Speech is subjective to personal or group think. Censorship is censorship.
I don't think it's all that subjective. If the board says: "Legalize abortion, protect women", it may offend pro-life supporters, but it's no hate speech at all. If it says "Pro-life supporters oppress women", well it may qualify as hate speech if it's a statement that's unsupported by facts, but it's still a stretch. If it says "The only good Pro-Life supporter is a dead one", then it really is hate speech. In America, legal tradition is usually lenient towards hate speech because it sees it from the practical consequences perspective, and hence protects "free speech" to enormous lengths. If it didn't result in violence, it wasn't hate speech, end of case. (That may well start to change after the attack on the Capitol). But it Europe, especially after fascism and WWII, we tend to see hate speech much more from the potential damage it can cause to society, and thus are much less lenient, even if at times over-protective of people's sensibilities.
But anyways this thread is about the question whether the right to not be exposed to ideas I don't like or agree with does exist or should exist?