> Couldn't the alert instead be rendered as an overlay to the page itself, be modal to just that tab, and not disable the browser Chrome?

Firefox and Safari do exactly that.

> The existing policies (don't allow this page to create more dialogs -> page instantly redirects to itself to reset the flag) aren't doing a good enough job.

Does that work? I've never seen any pages that circumvent alert blocking like that, and I'd have imagined that any sensible implementation would disable alerts for the entire session (or at least a few minutes).