How about after load, that tab automatically goes completely offline. Users can manually do this in Chrome on a tab by tab basis by using developer console and setting Throttling to "Offline"

That sounds more promising. The site might be able to store data and then send it the next time the page is loaded. I think at the end of the day, a malicious dev could probably find a workaround to most implementations. Might just be better to vet out sites and use reputation to state they are truly offline.

I hear what you're saying, but I also believe this to be a solvable problem

Note that the "offline" mode in devtools doesn't kill any websocket connections. It may have other holes i'm not aware of also.

Or in the subdomain, using "DNS exfiltration": https://twitter.com/rsobers/status/1293539543115862016

That's an interesting exploit, thanks for sharing.