The point I'm making overall is that demanding there be no anonymity is asking everybody to give up an exceptionally valuable freedom, and what we're getting in return just ranges between nothing for the overwhelming majority of the population, to things with relatively easy solutions for the small percent of people that are affected.
You seem to think people should randomly trust any old LLC to be honest with the public. Why?
As for lying, this is why you have contracts, and in those contracts you have penalties above and beyond losses as penalties. Incidentally it would also generally be a crime. Fraudulent representation of ownership or identity would certainly be prosecutable as fraud and likely other charges. If any money changes hands, you're also setting the stage for wire fraud charges which are particularly nasty, and what really packs on the decades for basically every white collar conviction.
A non exhaustive list of examples: an LLC trying to do business with your town; an LLC filing permits for construction or some business venture; LLC bidding on public projects; an LLC filing for eminent domain privileges at the muni, State or Federal level.
Etc etc.
In general, hiding ownership of LLCs is not done for benign reasons.