I've been hunting for a solution for this for weeks. The Chrome team said they fixed it and locked their support thread frustratingly.
It seems to occur most often when I am switching from one app to Chrome with Chrome in full screen mode. I do not have the issue if I use Chrome in a window.
When the pinwheel does come up, it usually helps to navigate away and click on an empty desktop, then come back. Interestingly enough when I do that, I occasionally notice a "this program is recording your screen notification" in the status bar.
I actually gave up on Chrome because of this issue, but reinstalled it after noticing some of the other browsers lacking some of Chrome's unique features