Think I’ve posted here before regarding a reasonably fast fix - I used a set of buffing wheels on a drill with glass/metal polish. It was pretty fast. The buffing wheel kit I got had two cloth and one fiberous one. The fiberous one is fast but can scratch the glass. Probably best to be patient with the cloth wheels and use the polish. Yes, the middle is hardest.
Just focus on some small area and polish away, then move to the next small area. Good luck. My screen looks great, but doesn’t have anti-glare. I can’t say I’m missing it.