A bank won't be using a self-signed cert. So it will be presented with the "this page (attempts) to be secure" UI, including whatever scary warnings are needed if things seem broken. Also, you could show self-signed pages with a broken padlock somewhere, e.g. beside the HTTPS in the URI - or you could /not/ show the "https" part of the URI for self-signed, etc (Chromium replaces the protocol scheme of URIs with icons).
The key point is that self-signed shouldn't be /worse/ to use than no-security, given that self-signed definitely solves /some/ security problems.