On one hand I find it hard to believe they do not use it. It's not like it's buried in the analytics dashboard. It's a pretty prominent metric. However, I have gotten websites organically ranked that I never implemented an Analytics tag on.

They haven't said it, but they absolutely do it, or at least did about 7-8 years ago.

Source: used to chat with Google search engineers about web spam back in the day at conferences.

“Google Analytics is not used in search quality in any way for our rankings.” -Matt Cutts (ex-Google) (2010) https://www.youtube.com/watch?v=CgBw9tbAQhU

Ah yes, 13 year old information. Maybe it was true then, it is not now. Cutts had a known pattern of denying anything and everything about their ranking data until cornered (see his video with Harry Shum of Bing when they caught Bing stealing Google clicks), so I would recommend against using him as a source for almost anything these days.

Believe what you want, but apply some basic logic - why would they not use data they already have?

Yes, Google has said this. The use of telemetry from Google Chrome has factored into SERPs as far back as 2010 (perhaps farther but that's when I became aware of it).