>> his seems pretty straightforward to me. “And subject to the jurisdiction thereof” <<
An ambassador and his family are not subject to USA jurisdiction, as noted by Senator Howard. They have diplomatic immunity. So if an ambassador commits murder in the USA, we can’t prosecute him. We can only expel him.
By contrast, if an illegal alien commits murder, he IS subject to USA jurisdiction. We can arrest, try and put him to death. If you think the latter step isn’t a fundamental aspect of “jurisdiction,” then you and I would seem to be living in parallel universes, meaning moreover that a rational discussion of the matter is probably impossible.
Assuming, of course, that the murder occurs on US soil.
A US citizen, OTOH, can be charged with US crimes, and has to follow (some) US laws, regardless of where on the planet the US citizen happens to be. Tax laws, registration for the draft, and some sex laws have that property; so the US citizen is "under more jurisdiction" (for want of a better way to say it) than the illegal alien is.
And if a diplomat can be expelled, does not that make the diplomat "subject to the jurisdiction" too? Albeit just a little bit.
I think there are cases where diplomatic immunity has been waived after the fact, by the home country, so that a murder charge can proceed. Ex-Diplomat Gets 7 Years for Death of Teen in Crash - latimes - December 20, 1997
At any rate, the issue as neat, tidy and simple as many people think.