Neither the U.S. Constitution, nor any law prohibits discrimination based upon national origin in immigration laws. Indeed, U.S. immigration laws and treaties require discrimination based upon national origin in many cases. The equal protection clause of the 14th Amendment has not been interpreted to prohibit discrimination on this basis in immigration laws.
State laws that discriminate based upon citizenship or national origin are on much more fragile ground under the 14th Amendment, because immigration is exclusively a federal government function under the United States Constitution.
The Civil Rights Acts, which do prohibit discrimination based upon national origin, apply to public accommodations in the United States, employment of people who are legally permitted to work under immigration laws, and a variety of other matters, but a person who is not a U.S. citizen does not have a right to a visa, and immigration laws (which have been upheld as constitutional repeatedly), have always discriminated based upon national origin. The circumstances under which national origin discrimination are prohibited are listed at the link in the original post as:
- Public Accommodations
- Law Enforcement / Police Misconduct
Immigration laws have almost always historically made distinctions between nationals of different countries that seem unfair and will continue to do so. Similarly, citizens of some countries can visit the U.S. for short periods without a visitors visa, and others may not.
These issues are within the sole discretion of Congress and the President, and even immigration courts are Article I courts rather than Article III courts until the U.S. Court of Appeals provides a second layer of appeal from immigration decisions. The Supreme Court has called their authority in this area "plenary" (see Cole 2003 starting at page 384 citing e.g., Piallo v. Bell, 430 U.S. 787, 796 (1977) (quoting Mathews, 426 U.S. at 81-82); The Chinese Exclusion Case, 130 U.S. at 603-06.). As Cole (2003) states at 386 (footnotes replaced with inline citations):
[An] argument commonly heard as a rationale for affording noncitizens
less robust rights protection maintains that because noncitizens are
only "guests", Mathews, 426 U.S. at 80, who have "come at the Nation's
invitation,' Carlson, 342 U.S. at 534; Foley, 435 U.S. at 294, their
admission and continuing presence may be conditioned on whatever
constraints the government chooses to impose. As the Supreme Court
once put it, deportation "is simply a refusal by the Government to
harbor persons whom it does not want. Bugajewitz v. Adams, 228 U.S.
585, 592 (1913). If you don't like it, the argument goes, either don't
come, or get out. This argument seeks to transform what we generally
think of as inalienable rights into discretionary privileges that can
be granted or denied at will.
As Cole (2003) explains, not everyone agrees that this is how the law should be or what the constitution should protect, but it is a reasonably fair statement of the state of constitutional law regarding immigration, especially in the 20th century and before.
Almost all rights to persons who are not U.S. citizens to immigrate or visit the U.S. arise from federal immigration statutes, rather than any other source, and even then, for the most part, immigration officials have some residual authority to deny admission to the U.S. to people who otherwise have valid visas allowing them to enter the U.S.
Also, it is worth recalling that sometimes U.S. immigration law discriminates based upon "nationality" or "citizenship" rather than "national origin" per se. But, often a person is "chargeable" for visa purposes under U.S. law to the county where they were born rather than the country they are currently a citizen or national of (leading to true "national origin" discrimination), although exceptions apply when an immigrant is married to someone born in another country, and U.S. law doesn't generally look back to where your parents were born if you were born in a different country than your parents.