We the People of the United States, in Order to form a more perfect Union...

Not if we read the text:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

Extracting the relevant bit:

Congress shall make no law... abridging the freedom of speech

Illegal immigrants and visitors "have" free speech because no laws may be enacted against it, period.

Illegal immigrants and visitors have the right to assembly because no laws may be enacted against people assembling, not just citizens assembling.

And perhaps more significantly, let's look at the 14th amendment (which I really should have started with in the first place):

>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. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

Here there is a clear distinction between citizen and person

Yes, there is. There is also a clear distinction between which persons it applies to and which it doesn't;

> ... nor deny to any person within its jurisdiction the equal protection of the laws

There is a clue though:

We the People of the United States, in Order to form a more perfect Union...