Forgot Password typical use case

This is "inspired" (a crosspost?) by this question I asked on StackOverflow Meta: to my surprise it has been highly downvoted, so I'm asking here to you experts, if my "feel" is sharable or not.

Basically, this is what you see when you want to login to StackOverflow:

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And this the page you are redirected on when you click on the "Forgot password" link, regardless of if and what you have typed in the Email field in the first page

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My suggestion is that "Forgot password?" should be clickable only if something has been typed into the email address field, and should immediately send a recovery email to that address: no need for the second page at all.

So, the question(s):

  1. My feeling is that asking twice the email address is wrong. Am I right?
  2. I believe the current implementation does not reflect the typical use case, that (I believe) is: try a bunch of passwords, give up, click on the link. Is it so?
  3. Can the current implementation be improved?
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