You'd have to physically change the action taken on that User password page in the Site Manager because the action is physically changing the password, which is why it sends the actual password to the user and not a link. The link triggers them to reset it, that button click has already reset it. Take a look at the physical file that does that work here: /CMSModules/Membership/Pages/Users/User_Edit_Password.aspx.
You could create your own version of it and have an additional button or refactor that other button and have it do what you want. In fact, you could create a webpart that does this now with v8 and simply change the UI structure in the User app/module to use a standard page template with your webpart.