No, you are misinterpreting and misapplying the "law"
Just so you get a grasp; there was no "immigration laws" , so therefore it would be nonsenscial to apply any regulation to the original intent of the framers; the founders would have never dreamt of such a law

There is no immigration law persay in the constitution. However there are federal laws governing immigration, and what is required for obtaining, visas, citizenship, work authorization for foreign workers, etc.
But most are not enforced, as with any law they are only as good as the enforcement.
Also with any law that is written, there is another law that cotradicts it. Sometimes the cotradiction is within the law itself.
The immigration laws has so many loopholes, and contradictions it's almost a joke, and very difficult to enforce in some areas.

The reason that immigration is REGULATED (critical word) , and has only evolved post WW1, was for two reasons, syping and sabbotage; and the organized criminal element that neccessitated the evolution of "screening process" and to have a legal tool ( deportation) to deal with "undesirable aliens"; that we were unable to convict "beyond a reasonable doubt".
America has a long tradition of encouraging hard working people from all countries to come share the American dream; ans they never had an easy time, with each wave of specific immigration , those people became the working underclass, french, irish, chinese, eastern europeansincluding the european jewery( ashkenazi) , italian, southeast asia, and now hispanic. even now the shidft from hispanic (as a generation work there way out of the underclass) towards immigrants from various countries in Africa and the middle east.

No one wants the criminals from other countries, as we have plenty of our own. But by making immigration to rigid and difficult, we are excluding pools of people whose work ethic and brain pool willl better our county through intellectual diversity. As we all know, when laws become too strict, the honest people are the only ones that obey the rule of law; the criminals get here anyway. THAT should be our focus, amnesty for the hardworking HONEST people that have overstayed there "tourist visas", and channel our resources to eliminate criminals from all other countries Russian, Hispainc , Asian, etc.

I think Native American Indians are about the only people who weren't illegal immigrants. The rest of us just showed up and claimed the land as our own. There is no constitutional law for immigration. America is a nation of immigrants. For most Americans, immigration is somewhere in their family history. Then, as now, immigrants were looking for a better life.

I like the second half of your question. It seems that the federal government would have the right to sue a state for issuing driver's licenses to illegals but it hasn't yet.