I know a (thankfully small) handful of cases in which Americans' kids were arrested in a friendly country. The State Department provided invaluable support. (Recommendations on legal counsel, prudent next steps, relevant authorities' phone numbers...nothing nefarious, just context and support.)
I am under no illusion that I, with my resources, would get that sort of access. But it exists, and probably benefits U.S. international business concerns. Since this article concerns itself with rich overseas Americans, I think it's relevant.
One thing the US has is more bargaining power to force early release of some people accused of crimes (like Liangelo Ball in China, Anne Sacoolas in UK). That's significant leverage, but again: most people will never find themselves in that situation. But even then it's not like other countries will not imprison you if you're American.
Not many millionaires are getting arrested in general, due to the availability of lawyers and so on.
The US support might be very relevant for a war reported captured by ISIS.
But most millionaires are not going to be in those locations and/or able to afford good private security.
It could even be said Americans are more at risk in those countries for being American and no other reason. So for people outside of that there won’t be any benefit.