Is that really true? I am not a US citizen but I have an account with a US broker. I just need to file a single form once in a while declaring my foreign tax status and that's it. This can hardly be the reason why it is absolutely necessary to have offshore holdings.
(Of course, I have to pay taxes at home for any profits and there are supposed to be mechanisms that ensure I do...)
If you deal in volatile markets with questionable rule of the law and weak property rights (e.g. Russia or much of the Middle East), then the ownership and its transfer is often structured with offshore companies.
The leak however suggests the legit offshore cases are drop in the bucket in this overall money laundering business. Kind of like with file sharing, many would argue we need it primarily to get our latest Ubuntu ISOs and other legit uses, but we all know it's a lie.
I believe it depends on the nature of the investment and your relationship with the US. For example I believe in your situation you'd have to pay US taxes on dividends from US companies.
(Of course, I have to pay taxes at home for any profits and there are supposed to be mechanisms that ensure I do...)