For several hundred years, undocument immigrants were deported only if they were convicted of a serious crime or ended up as public charges (on welfare as we say today). Most of our ancestors entered America as undocumented aliens under these rules. The reality, if that if current laws were in effect in the past, very few American Jews would have been allowed to immigrate (and the decision back then to let our ancestors come was very controversial). American was built on the assumption that anyone wanting to come here and work hard to build a life for their family was welcome – that’s the American way.
The 14th amendment was intended to be broader than making the ex-slave into full citizens. In the debate, it was asked to the effect, do you really want us to be stuck with Jews and Irish as well – and the answer was “yes”. Since then, “jus soli” has become deeply ingrained in the American psyche – anyone suggesting someone born in the US is a “foreigner” is considered by most people to be un-American.
There always have been “nativists” opposed to “new immigrants” (e.g. anyone whose ancestors came after the revolution). Please note that of the leading Republicans, only Bush and Carson can trace their ancestry back that far. Very few people reading YWN can. The nativists were politically important in the 1840s (too many Jews and Irish coming), but never were close to winning.