Chrome's screen sharing function is the best among all the browsers. However, you might still run into some problems.

Share Button in Grayed Out

This really isn't a problem with Chrome, but that it's not intuitive that you have to select one of your screens/tabs/applications first:

Start screen share

Click on one of the options in the tabs: Your Entire Screen, Application Window, or Chrome Tab
Now, select one of your screens, application windows, or Chrome tab, and THEN the Share button will light up.

On a Mac: when choosing the screen, window to share, the thumbnails to choose from are grayed out

This is likely a permissions issue. Try the following:

Bring up System Preferences on your Mac
Go to Security & Privacy
Click on the Privacy tab
Find "Screen Recording" on the left.
On the right, see if "Google Chrome" is checked.
If not checked, check it! You may have to unlock this panel to make changes!
Restart Chrome.

No audio when sharing a video from a Chrome tab? See this article.
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