Apparently not. Our billionaires continue to offshore their cash to countries like Switzerland.
And the fact that our companies continue to do exactly the same thing, also proves this out. There's no major international bank that won't take on lucrative accounts with our mega corporations.
What you're talking about is wishful thinking at this point.
Effectively 'hiding' money offshore though opaque company ownership structures is possible when you have jurisdictions where even asking who owns the company could result in jail-time (see 5.1b of the Caymans confidential relationship law [1]). When you don't know who owns what, it becomes difficult to selectively 'turn people away'.
That's a really really far cry from what you said in grand-grand-parent. From "there isn't a major international bank that will turn away American billionaires" to a numbered account in Switzerland, a country that's fairly unique in their bank secrecy legislation and tradition.