In other words, I want only the account/profile owner to see the Edit field (/user/[nid]/edit) on their own page and to be able to edit his/her profile info. How to do this?
Create the block without the edit link. Then implement function hook_views_pre_render in your theme.
This hook is called right before the render process. The query has been executed, and the pre_render() phase has already happened for handlers, so all data should be available.
Adding output to the view can be accomplished by placing text on $view->attachment_before and $view->attachment_after. Altering the content can be achieved by editing the items of $view->result.
This hook can be utilized by themes.
Within this you do a conditional check to see if currently logged in is the same as profile owner, then if so add the link to
I think you would get quite a lot of answers on this question as there are so many ways to do it.
If you want to do it without coding I think you might get it right with Views arguments in filters installed. You create the link field in an attachment to the block using the currently logged in user as contextual filter, then (via installed module referenced) do the conditional check in your filters section of views on this attachment.