To be clear, though, I don't think non-citizens are voting en-masse. My concern is that if you aren't even verifying they're citizens, you probably aren't really verifying that they are a real and unique person that isn't already registered.
Honestly I think if we actually wanted secure elections, we'd start with the red teaming and go from there. The signal to noise ratio of fraud is too meaningless to resolve without tightening up rules, which the results of the red teaming would give you the political capital to do.