I don’t think the devs would be against implementing something like this, but implementing something like this would present a major technical hurdle more than anything.
Rabb.it (now defunct) had this exactly, but it used virtual machines running a web browser, which wouldn’t be a viable option. Having the host screen share a window would also result in things being perfectly synced, but would have the same issues (plus, you’re hosting a condo and streaming video, so performance would also be an issue).
You could also try having the browser run on each client and synchronize the mouse cursor and open webpage, but any differences in page loading (buffering, ad placement, etc.) would throw things off entirely. Same goes for any website without a synchronization API, since any buffering would make things no longer synced.
This is also ignoring the current web browser’s codec issues that prevent livestreams from working properly. Syncing raw video files (rather than entire webpages) would be a lot more viable (and is basically what the first option boils down to), but I’m sure that has its own share of issues.