Mostly true, but at least in San Francisco, non-citizens can vote in school board elections. Here are the rules (below), and I’d like to note, there is NO requirement for an applicant to show documentation. (SF is a sanctuary city, as well.)
"To register as a non-citizen voter for the next Board of Education (School Board) election, you must be all of the following:
(1) A San Francisco resident who intends to remain so until the next Board of Education election;
(2) The parent, legal guardian, or caregiver of a child living in the San Francisco Unified School District (i.e. living in San Francisco);
(3) At least 18 years of age on Election Day (the next such date is November 5, 2024); and
(4) Not disqualified from voting under state law for imprisonment or mental incompetence.
https://www.sf.gov/non-citizen-voting-rights-local-board-education-elections