How does this work from a technical perspective? If I forward mail to this service instead of clicking the 'unsubscribe' link, then how do they 'unsubscribe' me on my behalf? In order to complete the unsubscription process, the newsletter owner needs to receive an email (usually generated automatically by the mail client when clicking the 'unsubscribe link') that comes from the person being unsubscribed. If I outsource this process, this means that 'please-unsubscribe.com' will need to spoof emails from me on my behalf and send them the email specified in the List-Unsubscribe email header. If spoofed emails are not used and they come from 'please-unsubscribe.com' this won't be very actionable to the newsletter owner. From their perspective, they're now going to get 'unsubscribe' notifications that all come from an email at 'please-unsubscribe.com' which never signed up to the newsletter in the first place.
This all assumes that the newsletter unsubscription process is using the email based method instead of the link based method (which is a good assumption email-based is more common).