Would they really, though?
A lot of people say this, but I think one very important thing is being overlooked: should a billionaire leave the US, they'd also lose a significant amount of influence over US policy. Especially when their wealth is built significantly upon the stocks of major US companies, giving up that influence is a huge risk to their wealth.
Just think - not only would the billionaire's influence suddenly become stigmatized "foreign influence", but it would also become tainted by the fact that they fled the country to avoid taxes. Politicians on the opposing side of an issue the billionaire wants to fight for would have a field day pointing out both every time the billionaire attempted to influence any policy.