Perhaps it was both tongue-in-cheek and not. But if they're not doing anything "wrong" then what is there to fear? Nothing.
What will really happen is that we find out that there are millions of LLCs owned by a few corporate investment offices that are driving up the price of housing for my fellow citizens, actual human beings who I value far more than any legal fictions.
Anyway, just to paint maybe some more realistic threat models: Swatting. People suing rich people to get "go-away" money. Targeted robberies or abductions.
Privacy is a right, also for rich people. For any information that is private, I can find a good reason for it to not be private.