> They understand why they need to keep their bank passwords safe, and hence they can understand that they shouldn't enter it on someone else's website.
Not in my experience - people reuse passwords regularly, and are exceptionally vulnerable to phishing scams, as you should ask anybody working in corporate IT. This is why people actually working in security are very excited about U2F - a U2F-generated token is tied to a specific domain, so password reuse and phishing are both no longer as serious a problem, since a login to scammer.com no longer works as a login to mybank.com.
On top of that, people give their bank passwords to e.g. mint.com on a regular basis, when it's specifically asking for their bank password and they have no reason to trust that website.
> they understand why they shouldn't give their ID numbers to random strangers
Also not in my experience, and the basis of many scams.
People at large are terrible at security. We've been moving towards "automatic security" for a while for a reason. And that's before you get to the phenomena where critical thinking goes out the window as soon as something is on a computer instead of "in real life".
> Surely someone's already looked into this kind of an approach and what you're saying isn't just pure speculation?
Moxie Marlinspike's Convergence would be the closest thing, iirc. Whenever you access example.com, you ask a handful of user-selected notary servers to access example.com and tell you its TLS cert. If all of them match, you're not being MITMed. You don't have to trust any individual notary server, because it's all of them put together that provide the trust.
Of course, trying to figure out a good UX for "my Government and Google say it's fine, but Estonia and Facebook say it's not" in such a way that users aren't going to be severely inconvenienced by a malfunctioning/MITMed notary server but also aren't going to click on "I have no idea what this is just let me see the website" on a MITMed page is hard - probably impossible. Moxie has since given up on it.