it's not that woke people control the institutions - it's that companies see that they can run successful PR and marketing campaigns by espousing progressive values. Their boards don't give two shits about progressive values, but flying a rainbow flag during pride month doesn't require that they do anything, while gaining praise from liberals and criticism from conservatives, both of which are coverage/press.
In addition to this, white collar workers are going to align a little more left and it makes them feel better if the company isn't cheering for gays to be lynched or whatever. It's generally easier to hire for highly educated positions if you appear mildly woke, even if you do absolutely nothing to that end.
