After dicking around with the API for a while I found that that you can highlight/dim tabs using chrome.tabs.highlight running in a background script of an extension. Maybe I'll spend a little while tomorrow making a more complete version.
Yes it suspends the process, but doesn't visually indicate this by fading/sorting the tabs for you. At least, not the version I used a year ago. (edit: it does now!)
I have always felt like this method sounds good, but is just annoying to do. I want to login fast, usually with passwords that autofill from lastpass or similar app.
Maybe if the information on the site was sensitive it would be worth it, but this is too much for just any random app.
I've just never actually seen anyone else do it.For some reason, it feels 100x more complicated than checking my SMS app for an SMS code when in reality its not that different.