Had the same problem for a couple days. Not sure if I got lucky or if this actually did something...
A clue was in the Download section, which for Chrome now says:
"Features dependent on a binary component, such as sharing of login state with other browsers, will not function."
What I did was:
Everything was back to normal.
And now, half an hour later, suddenly it's working. And the button still says "Enable Native Messaging". #shrug
Unfortunately, that did not work for me. In fact, after restarting my browser and checking the extension again, the button still says "Enable Native Messaging". It did not change.😞