I have the exact same problem. For no apparent reason, the clipper stopped working. I log in, but the dot won't go away. When I click on the clipper button again, Safari opens a new tab to redirect me to Evernote so I log in again. It then realises I have already logged in and automatically closes the new tab. I'm going to cancel my subscription as this has rendered Evernote useless for me.
Same here, on Safari 10.0.2 the Evernote clipper 6.10.1 just stopped working.