... if I simply flatten the whole document to a "image style" PDF vs a "text style" PDF will it remain Accessible, specifically ADA compliant? I'm thinking of screen readers, since there would be no "text" to select, there would be nothing to parse/read/etc.
It would not be "Accessible" to vision-impaired people. As you've pointed out, "there would be nothing to parse/read/etc". There may be screen readers that use OCR. However, since the conversion to an image format would have been intentional, with the knowledge that it would hinder ordinary screen readers from operating, you might want to contact your lawyers to give them a heads up. (ADA tends to be very litigious friendly. People lose cases even when violations are unintentional.)
I've been tasked with creating a PDF file of some student records that can't be edited, copy/pasted, etc.
If a user has permission to view a document, it's possible to modify them so that editing is possible. Conversion to an image format doesn't stop anyone. It only slows them down. Consider how quickly some paper books were digitized before ebooks were commonly available.
I know about digitally signing, etc.
That's pretty much the only way you can ensure a document will reach its destination unmodified. Someone could still change it, but they won't be able to sign it.