There IS a requirement to prove one is a citizen to get to vote under motor-voter laws. Note which states are exempt under the law:
https://en.wikipedia.org/wiki/National_Voter_Registration_Act_of_1993#Scope_and_exemption
Note which states require ID at the polls and which ones which require ID to register and what forms are required.
https://ballotpedia.org/Voter_identification_laws_by_state#tab=Details_by_state
The idea that there is a widespread ability for folks to register and then vote illegally is not founded in fact.