I think a lot of people don't realize that bank account privacy is just 1 of 20 distinctive qualities of Swiss institutions and regulatory structure.
Switzerland is attractive because its not just shell companies and PO boxes, compared to other nations that try to offer similar nominal regulations. Instead it has a vibrant regulatory system with competition amongst each state and no expensive federal government. As an example Google pays the same as it does in Silicon Valley, with employees being only subject state level tax. The state of Zug is a fintech hub with an income tax rate of 7%. Territorial only. The Swiss franc is currently pegged 1:1 with USD, having ditched Euro peg almost 5 years ago.
The regulators and public sector representatives are very accessible because everything happens at the sparsely populated state level, and they all operate under the swiss brand which includes access to the global financial system.
Switzerland is better thought of as a loose collection of sovereign states with a small national government. The concept may seem familiar but recognize how small all of Switzerland is and that has factored into why the national government never grew arbitrarily large to maintain cohesion and assume its own intrusive identity, and why direct representation has been possible there for almost 200 years (they did represenative democracy before that, like the US, but figured out how to remove it since the flaws were obvious and they're smaller)
Outside of finance and regulatory structure, Americans may also find the vibrant gun culture and visible military to be refreshing and familiar. A stark contrast to the ideals and vocal opinions that neighboring Europeans would have about guns as soon as they find out you're American, possibly unaware of what Switzerland offers and has offered for a very long time.