Let me get this Google tweet out of the way first. Gary Illyes from Google tweeted this back in 2015: "we don't use analytics/bounce rate in search ranking." And Matt Cutts, the main man for Googles search algorithm, said something along the same lines. So that's pretty clear, right?
I'm not saying that the bounce rate of a website isn't a direct ranking factor. Also, Google definitely does not need analytics to show how long a user has been on a website. What I do believe is that in some indirect way, bounce rate is a factor when it comes to your rankings.
I'm not saying that a high bounce rate won't be the main thing to kick you from the first page. I just feel that it's one in a sea of factors that help or hurt your rankings.
Now click through rates can be altered and gamed pretty easily. If I put up an ad that said "You get a free cookie by clicking here" you bet I'll be getting a lot of people clicking through to see if they actually get a free cookie. But when they get to the page and there's no cookie, they're going to leave. And this will increase your bounce rate.
So I believe that Google does track your time on site just to make sure your links are genuine and aren't just click bait.
Another question I see a lot is when people ask "Will higher rankings increase my conversion rates?" Well, it's extremely difficult to figure that out exactly. You could be on page one rank one and get 100 views and 10 sales while your competitor is on page 2 rank 10 and getting 10 views and 1 sale. The conversions are the same, there's just less sales on page 2 obviously.
So in the end, try to get your users to stay on site and click through a couple pages. Make your pages user friendly and easy to navigate. I say this because a good bounce rate and time on site can be a small factor in your rankings in the search engines