What else would you call DACA, sanctuary cities, or the proposed amnesty for illegal immigrants, then? Eventually, they or their children become citizens, which is little different from the policy you claim literally no one is proposing.
==Eventually, they or their children become citizens, which is little different from the policy you claim literally no one is proposing.==
Literally none of those are "giving citizenship to everyone who wants it". To take your eventually example to the next step, if their children's children have kids on US soil, they will automatically be US citizens. By your logic, the existing US Constitution is "little different" from the same policy.
Which is why they apply for asylum, a process enshrined in the US Code.
8 U.S. Code § 1158. Asylum
Any alien who is physically present in the United States or who arrives in the United States (whether or not at a designated port of arrival and including an alien who is brought to the United States after having been interdicted in international or United States waters), irrespective of such alien’s status, may apply for asylum in accordance with this section or, where applicable, section 1225(b) of this title.
