I see that you first posted this issue around 12 months ago (FEB 18). Fast forward 12 months later to FEB 19 and it is still an issue. This feature to 'Clear on exit' in the Chrome/Privacy and security/Content settings/Cookies area is not working. Cookies are still retained for any added sites after Chrome restarts.

I have a feeling that this has something to do with not interfering with subscription paywalls, but I digress...

There is a manual workaround for this. While not fully automated (or ideal!), it barely takes a few seconds to do.

One-off step In the Chrome browser URL bar, directly input the path 'chrome://settings/siteData' then save a bookmark to it. If - like me - you don't mind having the Chrome bookmarks bar always visible, it gives you instant access to manually clear cookies by site.

At the start of every session (only if you will be visiting the site in question)

Click on the bookmark Search for the cookie(s) in question Click on the trash icon to clear them

Note: you don't need to restart the browser after the above steps.

Hope this helps.