I know a lot of conservative folks. I think people on the left dramatically overestimate how much control and influence Fox News has on the average conservative. Do they have influence? Certainly. But Fox News has been the trendy go-to explanation on the left for why conservatives think or behave a certain way. I also think a lot of people on the left don't yet grasp just how many conservatives are enraged at Fox for 'betraying' Donald Trump. I don't want to ramble, but all-in-all, I don't think the change of tone at Fox will have a significant impact on vaccine hesitancy/intransigence.
This is an interesting instance in which to measure that. It's hard to tease out the cause-and-effect here. You get the same thing with politicians: are they saying something to convince their constituents, or are they saying it because their constituents are already saying it and they know it's what they want to hear?
My suspicion is that it's some of each, and I'd bet that the numbers in red districts will rise some, but not a lot. Proportional to the ~2 million Fox News viewers, out of ~80 million people who voted Republican last time. Not that all Fox News viewers will rush out to be vaccinated, but that [say] 25% of them will and influence an equivalent number of non-viewers.
https://www.vanityfair.com/news/2021/07/whats-behind-fox-new...
I don't know the reaction among Fox News' viewers. It'll be interesting to see if vaccination rates tick up markedly in red districts.