Yes, states could allow aliens to vote for President. As ohwilleke says, the Constitution gives the states control over who can vote. In fact, for much of our history, many states allowed aliens to vote. To the extent that 18 U.S.C. § 611, which forbids aliens from voting for President, contradicts that power, it is unconstitutional.
If 18 U.S.C. § 611 is so obviously unconstitutional, why is it still on the books? Mostly because 18 U.S.C. § 611 is purely symbolic. It literally has had no effect on who can vote in the US. As noted above, throughout the 19th and early 20th century, many states allowed aliens to vote. But with the turn against immigration in the early 20th century, states that had allowed aliens to vote explicitly revoked that privilege. The last state to prohibit aliens from voting was Arkansas, in 1926. Thus, for almost a century, every state has prohibited aliens from voting in state elections, so no voter was affected by the passage of 18 U.S.C. § 611. (For a survey of the history of alien suffrage, see here.)
Voting is a state matter:
It is widely agreed that the Qualifications Clause, Art I, § 2.1 of the Constitution gives states control over who votes in federal elections, (subject, of course, to other constitutional requirements, such the 15th, 19th and 26th Amendments). The clause says
the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature.
In other words, anyone who is qualified to vote for the largest branch of the state legislature is qualified to vote for President. (The original text only talked about elections to the House because Senators were chosen by state legislatures, not voters. When this was changed by 17th Amendment, that Amendment repeated this qualification for voters.)(For more extended, but reader friendly, discussion of these issues see here.
People sometimes point to the "Elections Clause," Art. I, § 4.1, as giving Congress power over voter qualifications. That clause gives Congress the power to regulate the "Times, Places and Manners of holding elections." They read, "Manners of holding elections” to include the qualifications for voting.
This reading of the Elections Clause is inconsistent with several of the sources of constitutional authority Americans typically consult, including the text, the intent of the Founder and precedent. All of these support the “Qualifications Clause” reading of the Constitution.
1) Constitutional Text: Reading the Elections Clause as giving Congress control over voter qualifications, renders the Qualifications Clause almost completely meaningless. Yet everyone agrees the Constitution should be read so that all its parts mean something – no part should be rendered superfluous.
2) Intent of the Founders: When the Founders debated voting qualifications, they were clearly worried that politicians would manipulate the electoral rules to favor them. The Founders believed the Qualifications Clause kept both national and state politicians from stacking the electoral deck. First, by giving the power of deciding who could vote to the states, the Clause prevented national politicians from choosing their own electorate. Secondly, by requiring the states to use the same qualifications for voters in state and national elections, the Clause prevented state politicians from manipulating the rules in national elections.
As for the Electoral Clause, in Federalist 60, Hamilton was clear who controlled voter qualifications: “The qualifications of the persons who may choose or be chosen…are unalterable by the [federal] legislature…”
3) Precedent and practice: For most of our history, people acted consistently with Hamilton’s statement. They tolerated a wide variation in who was allowed to vote across states. As noted above, many of those states specifically allowed aliens to vote. (And, starting in 1787 with the passage of the Northwest Ordinance, Congress passed a series of laws allowing aliens to vote in territories.) When Americans did make nationwide changes to voting qualifications, they did so by amending the Constitution, as when they prohibited discrimination in voting by race, sex or age.
There are only a handful of Supreme Court decisions involving federal control over voting qualifications. In 1970, in Oregon v. Mitchell, the Supreme Court upheld a federal law changing the voting age to 18. The opinion of the Court, written by Justice Black, relied on the Elections Clause to say that Congress had the power to change voting requirements. None of the other Justices shared Black's views on the Elections Clause. Instead, they agreed with Justice Harlan, who said, “nothing” in the Constitution, “lends itself to the view that voting qualifications in federal elections are to be set by Congress.” (These Justices used the Equal Protection Clause as the basis for federal control over voter qualifications.)
Since then, no Justice has followed Justice Black. Instead, they have followed Justice Harlan. For example, in 2013, in Arizona v. Inter Tribal Council Justice Scalia, (in an opinion joined by Breyer, Ginsburg, Kagan, Kennedy, Roberts, and Sotomayor) said,
“the Elections Clause empowers Congress to regulate how federal elections are held, but not who may vote in them.”
From this evidence, I’d say you were right to be puzzled about the constitutional authority for the IIRIRA. A federal law preventing aliens from voting in federal elections undermines the text of the constitution by rendering the Qualifications Clause superfluous, goes against the clear intent of the Founders, and is inconsistent with 200+ years of history and precedent. It only survives because it has no practical effect on voting in the US. Since this situation seems unlikely to change soon, it seems likely the statute will remain on the books to confuse Americans for the foreseeable future.
Edited to add:
Could states count alien votes under the IIRIRA?
No: Under the IIRIRA, there can't be any alien votes to count! § 611(a)(3) of the IIRIRA says aliens can only vote in an election if they do not have "an opportunity to vote for a candidate for any one or more of such Federal offices."
You ask whether "non-vote" methods of measuring alien's opinions, such as surveys, are legal under the IIRIRA. These clearly violate the purpose of the IIRIRA, which is to prevent aliens' opinions from influencing the outcome of federal elections. But it's not clear they violate the IIRIRA as written. They only violate the IIRIRA if you are willing to apply the "familiar rule" that
a thing may be within the letter of the statute and yet not within the statute, because not within its spirit nor within the intention of its makers.
But in real life, these questions are irrelevant. As a practical matter, they would never come up. If a state really wanted to allow aliens to vote, it would challenge the constitutionality of the IIRIRA. That no state has done this means no state politicians want to be seen advocating for alien voting. The IIRIRA is like the proverbial law "requiring you to breathe:" You follow it, not b/c it's the law, but because that's what you were going to do anyway.