Well, based on what I found on Wikipedia
'Section 1 of the Fourteenth Amendment to the United States Constitution provides that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.'
Would indicate the Wastelanders born on American soil are by constitution American Citizens.
And as your quote proves - the Consitution can be changed - Even Amendments.
My personal theory to the structure of the enclave looks like this (I have posted something similar in the past, so if this looks familiar, thats why). Please note this is a personal theory, and is not supported by any canon sources; however it is also not contradicted by any canon sources that I know of, and can explain a few holes.
The President is elected, as per the consittuion by the Electoral College. The states choose the electoral college.
The US senate is appointed again by the states, in recent times this is by vote, but traditionaly this was done by the states governments.
I believe that the Power of the states has been transfered to the commonweath "governors" via emergency legislation (such as your classic Enabling act). Such emergency legilsation isn't unprecedented even in the US - Abraham Lincoln for example suspended Habeus Corpus. I know in the UK provisions exist(ed) for local Authorities to assume the powers of central government should central government become unable to function after a nuclear war.
As such, I believe that the "True Power" of the Enclave is in a "Commonweath Council" - each member via emergency legislation passed before the bombs dropped in their respective commonweaths has the power of the states and commonweaths "For the duration of the emergency". They appoint the president (as per the consitution), and senators.
If the consitution doesn't work for them - they change it. 2/3rds of the states can porpose an amendment, and 3/4 are required to pass it. As such, the Enclave can pretty much do anything it likes and it can remain consitutional - as long as 3/4 of this council agree.
This explains why in this case, wastelanders can be stripped of US citizenship - the 14th amendment has been repealed.
It explains why Eden was president - The appropriate people did decide he should be president, just like he said - the Electoral college, as appointed by the "Commonwealth Council", elected him.