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> Verifying their e-mail address would be useless as attacker would already know the e-mail.

How?

Everything you said is true for the implementation that was listed, but my point was short URLs for password reset aren't always bad, if other mitigations are in place, which should be in place anyway (rate limiting requests for password reset URLs and requiring verification of the email address).

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The attacker would start out with a targeted user's email or login to the site. A personal email address usually is usually public. Start the password recovery process. Use a botnet to try different shortened links. A rate limit on password reset for an account would help if the attacker had low probability of success before the reset link expired, but the attacker can cycle between multiple target accounts.

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