Voter ID in Presidential elections, and the Constitution

One troubling question has been nagging in the back of my head for quite some time: If the Constitution leaves it to the individual states to select the method of choosing Presidential electors, and they could, theoretically, choose outlandish methods like a paper airplane distance flight contest, a cow plop contest, a beauty contest, a net worth contest, etc., why can't the individual states decide to base the selection of Presidential electors on an election day poll of eligible voters who present ID?

Originally, the founders imagined that the state legislatures would appoint the electirs, among other options, and I am not a big fan of selecting Presidential electors by popular election, but if that is the method chosen by a state, why can that state not limit the voters in that election to those who present ID? Or, for that matter, to those who can demonstrate that they pay more in taxes than they receive in "benefits." Or those who own land?

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Voter ID in Presidential elections, and the Constitution