Wouldn't work well. Way too many of these companies key employees are not US citizens and many aren't in the USA at all.
Google, for example, has a large security team in Switzerland, with quite a few German and British employees. The NSA sees itself as a military organisation, it is bound by military rules.
> The NSA sees itself as a military organisation, it is bound by military rules.
What rules would that be? In the military, actively seeking (and using) information you have no right/classification to see is a serious offence. According to articles I've read, not a single NSA employee was disciplined for e.g. spying on their SOs or Exs.
Also: If the NSA doesn't have Swiss and German citizens working for it, it's not a very good intelligence agency. And we know for a fact that it is, at least as far as reach is concerned.
Why would that matter? Intelligence agencies have turned foreign nationals before. And in the case of a US/British cross-over, those intelligence agencies have intelligence sharing agreements.
Google, for example, has a large security team in Switzerland, with quite a few German and British employees. The NSA sees itself as a military organisation, it is bound by military rules.