If owners don't treat the LLC as a separate business, a court might decide that the LLC doesn't really exist and find that its owners are really doing business as individuals who are personally liable for their acts.
I have no criticisms of Nolo, but "the truly wealthy" hire lawyers. Those professionals are skilled at determining whether to set up partnerships, C-corps, LLCs, whatever. Besides we're talking about anonymity for anonymity's sake (well, also tax minimization and probably lots of other rich people priorities we peons can't imagine), not for dodging creditors. If the principal is poor enough to need to dodge creditors, probably the whole house of paper collapses.
(Probably also the lawyers are the first to know, so they'll get paid up front and be long gone at the time of collapse.)
Edit: maybe it’s not illegal, just a bad idea:
https://www.nolo.com/legal-encyclopedia/llc-basics-30163.htm...
If owners don't treat the LLC as a separate business, a court might decide that the LLC doesn't really exist and find that its owners are really doing business as individuals who are personally liable for their acts.