Doesn't heartbleed let an attacker hijack the keys? Then they can man-in-the-middle the connection, and spoof the file so that the website appear to be not vulnerable to the attack.
I'm not saying what you propose is impossible; I'm just saying it's quite unlikely. Security is all about layers. You don't throw away a layer that could be effective at increasing security in the majority of cases just because there are some cases where it wouldn't work.
This new layer will make the security better for the average user, but it will enable attackers to make a website look falsely patched in a MITM attack.
(That leaves the question of how useful it is to users; I guess it is nice for a service like lastpass, but the truly paranoid probably aren't relying on cues from services)