You yourself said there is no requirement to prove citizenship to register to vote. There are 50 million non-citizens in the country, legal and illegal.
I agree with you that "most" are smart enough not to register, but if only 2% register, that's a million people. And I would wager the number of non-citizens unaware they are not eligible to vote is far higher than you think.
The fact that people do not have to prove citizenship to vote is ridiculous. And your article you linked to was factually wrong when it said SSNs and driver's license prove citizenship.