You become woke by accepting and learning about reality. Colleges teach people about reality. The right wing is distinctly anti-intellectual because they depend on ignorance to maintain support from the moderate right.

I started out far-right because I was indoctrinated that way. Then I got an education thinking I was already smarter than everyone. Then I started to see through the lies I had been raised with. Lies about non-white people. Lies about gay people. Lies about colonialism. Lies about industrialization. Lies about other countries. Lies about leftists. Then I woke up.

half of America thinks the public school system is there to turn kids gay and trans

The latter. (Most) universities admit students who exhibit rationality, open-mindedness, curiosity, and a readiness to change one's mind in the face of new evidence. None of these traits are characteristic of conservatives, therefore academia is left-leaning.