Let's hope this doesn't get picked up by the (corporate) masses... the last thing I want is my browser offering personal TLS certificates to every server I visit as some kind of identity verification or fingerprint/tracking.
It's bad enough that ssh does this by default with all your keys.
Client TLS is rather unusable on the Internet by a typical random end user visiting a random public site, so that should at least keep the specific scenario you describe at bay.
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This is most probably where it's going in less than a year. The recent campaign "Safer with Google" in Chrome hints to this.