Even just basic hyperlinking would have to be heavily restricted.
For example...
1. Cross origin hyperlinks would have to be completely disabled
2. Same-origin URLs would need all url information (query params, path, etc) to remain completely client-side. The only thing transmitted to the server would be the base domain.
Post restriction there is no such thing as allowing network requests (same origin or cross origin) without opening a side channel which can work around said restriction (e.g. timing). However pre restricted mode there is no reason to restrict anything be it same origin or cross origin.
Pre-restriction would have to disallow running javascript in that case. Otherwise an app could save data to localstorage, refresh the page programmatically (or wait for the user), and then transmit.
Or rather have the browser treat the no-network session period with private window logic where the session is deleted if the no-network browsing session ends or dies (e.g. incognito puts a virtual session data container in RAM). That would even allow the app data (prior to triggering the mode) to stay cached without risking side channels with future resource loads.
The whole premise was to return to the pre-web 2.0 days but have the option for a user to enable modern web app capabilities. Which your idea would be a step towards.
For example...
1. Cross origin hyperlinks would have to be completely disabled 2. Same-origin URLs would need all url information (query params, path, etc) to remain completely client-side. The only thing transmitted to the server would be the base domain.
It's interesting to think about, at least.