Re-install OS X on an iMac from 2011
This isn't really an answer right now, it's a selection of possibilities which need testing &amp;/or confirmation they've been tried &amp; failed.

While there was still an old OS on the machine,  Cmd ⌘   R   should have run Recovery which would give you the opportunity to re-install the current OS, providing you could supply the Apple ID credentials initially used to 'purchase' the OS on that machine. Otherwise you would get the 'not available right now' issue.  

This is where the paradigm differs from MS Windows. With Windows the OS belongs to the machine &amp; you have to ring MS to move it to another, but not to reinstall on the same machine. With Apple the OS [&amp; many apps work this way too] is owned by the person &amp; can be freely moved or re-used, but not transferred to another owner.

Assuming the machine has at some point in the past had the necessary firmware update [that machine was first released  right on the cusp of this technology] then if either the Recovery partition fails [or has been erased] or you hit  Cmd ⌘   Opt ⌥   R  at boot, then it will go into Internet Recovery mode rather than local.
This will allow you, without any Apple ID sign in, to download &amp; install the OS that was first shipped with the machine - in this case 10.6.6.
From there it would be a two-step process to update to Sierra, as you would first need to instal El Capitan.  

A third alternative is to boot from the original CD/DVD - hold  C  at the boot chimes which will launch an older version of the setup process.

Your last alternative, as you suspected, is to take it to an Apple Store, who will do it for you.