No. They absolutely have not.
You can do a google books search on "natural born citizen" and turn up literally HUNDREDS of historical quotes that support the definition I and folks like Mr Rogers have given. The same search turns up virtually nothing to indicate anyone of any real authority ever relied on Vattel.
The Vattel claim is ABSOLUTE, FLAT-OUT NONSENSE. It's simply false. There is no evidence to support it.
It is NOT "merely pinin' for the fjords." It's passed on. It is no more. It has ceased to be. It's expired and gone to meet its maker.
It's a stiff. Bereft of life, it rests in peace. If a bunch of silly birthers hadn't nailed it to the perch it'd be pushing up the daisies.
Its processes are now 'istory. It's off the twig.
It's kicked the bucket, shuffled off its mortal coil, run down the curtain and joined the bleedin' choir invisible.
It is an EX-claim.
There's that claim again. Reminds me of Clinton's "I did NOT have sex with that woman."
I no longer have time to fool with this. I will drub you later.