>I'd say where the protection starts is that the board is yours. You can make it as open or restricted or curated or nonsensical as you wish. Other individuals can put up their own boards and they can display whatever they wish.
As mentioned in the article, Pruneyard Shopping Center v. Robins ruling rejected that logic. The government can regulate your conduct. The speech in your bulletin board is that of those who wrote it, not your's. Thus not 1A issue.
The underlying issue is if social media is a platform (like a bulletin board) or a publisher (like a newspaper).