It is POSSIBLE that Google consider bounce rate as one of the many factors to determine your ranking in the SERP. They are all for user experience. So it is highly possible that they design their algo to drop sites from the SERP that have high bounce rate. (Which is telling them that the sites are garbage and majority of the users that search for this particular keyword is not finding the contents that they are looking for in your site).
They could easily compare the bounce rate across the top 10 listings after X amount of visitors to those sites and adjust their rankings on the fly--pushing the sites with low bounce rate higher in the SERP and likewise, penalizing those with high BR. Hence the Google Dance along with many other unknown factors? God knows.
Their #1 concern has always been user experience. They would do anything to give a better user experience. The way i look at this, there is no reason why they wouldn't implement this in their SE, considering that they have plenty of datas on our websites.
Is is possible for them to know the bounce rate if we didn't install GA? Maybe..
However, I have plenty of thin websites that are MFA that have very high bounce rate (80%+) but my rankings are still staying strong at the top spots for months. Maybe the algo have not caught up with these websites since they're not competitive terms and might require more time to gather statistical significant datas? Who knows.
I'm pushing my luck with these low quality MFA sites. But I can easily make one and rank for another term that can last me another good 6 months. If you're concern with ranking your site well and maintaining it's top spots for a long haul, i'd install GA on the site and make amendments according to the statistic as much as Google would. It's either you make the changes to your sites for the users, or google will.
Of course these are all speculations. There is no right or wrong answers to the question. Just go along with Google and provide a better user experience and you will be fine.