Stepping back, and looking at the situation as a whole: the real problem is a lack of privacy laws. Banks, businesses and employers should be prohibited from sharing your personal information with third parties. I live in Switzerland, where this is the case. Even the government doesn't get this information. If the government thinks you're cheating on your taxes, they have to use warrants and follow the same procedure…
“Everything works just fine” It definitely worked great for a lot of dictators, tax cheats and the sort… I think Switzerland is a great example of why complete privacy isn’t fair on ordinary taxpayers - it allows the ultra-rich to hide what they owe
That said, yes, dictators and such were - and are - a problem. They aren't going to prosecute themselves, after all.
By the way, one of the top places unsavory types stash their cash is the US. FATCA is a one way street: US banks don't provide information on their international customers.