Unable to use a Time Machine backup

Question

I am getting very frustrated, as I cannot use a Time Machine backup the way I want to.
Short story, my Macbook Pro Retina (running Yosemite) is, and have for a while been, a litte buggy and slow, so I wsh to to a clean install of OSX, and use migration assistant to recover files and settings. I have tried to to a complete restore, but this hasn´t helped.
The problem is, the migration assistant can´t find the backups. When I restart with Cmd+R and prompt a restore from TimeMachine backup, it CAN find both mine, and other backups stored on the disk. But after I do a clean install, and get to the assistant, it CAN´T find any of the backups.
The backups are on a WD 2TB disk, connected via my Mac Mini.
I have tried creating another account on the fresh install, and verify disk and permissions. One time, that actually made the assistant find the backups, but I have never been able to recreate this. I did not try to copy the files on that occation, somehow.
I don´t understand how the two backup modes works differently. Complete restore cand find the files, assistant cannot.
IMPORTANT NOTE: The backups are created via a Mac Mini sharing the Time Machine disk. My backup is created as a disk file on Timemachine/Mybackup, and the Mac Mini backup is another level down on Timemachine/backups/Macmini. I don´t know if this has anythong to do with it.
On trying to recover, I have plugged the external drive directly into the MBP.
I have had to go back and to a complete restore every time, but I am not satisfied with this result.
I have tried to explain as good as I can, and will be very grateful for hints and tips.

Answer 1

A possible reason for backups not to be available in Migration Assistant is that the computer doesn't have a name, or the name is corrupt. Open the Sharing preference pane and check the name. If you can't do that, reset the PRAM. Credit to ASC member Csnote for this observation.
Otherwise, starting from a clean installation of OS X, set up a new administrator account and log in. Enter Time Machine and press the key combination shift-command-C. The front window will show all mounted volumes. All snapshots should now be accessible.* Select the one you want and navigate to your home folder (in the Users folder at the top level of the old startup volume.)
You should now be able to restore your user data. I suggest you do this in two stages. Quit all applications except the Finder before you begin.
Restore all the visible items at the top level of the home folder.
Hold down the option key and select Go ▹ Library from the Finder menu bar. Enter Time Machine and restore all items in the Library folder. Log out and log back in as soon as the restore is complete.
Any other invisible folders or files at the top level of the home folder that you want to preserve will have to be restored separately. For most users, that isn't necessary.
You may have to reinstall all third-party applications from scratch. That would be the only point of doing a clean installation.
You'll have another problem because you erased the startup volume. The next time you back up, Time Machine won't recognize any files as being the same as they were before, and will make a full copy of all files. There might not be enough space on one or more of the backup volumes for that. There are different ways of dealing with that situation, depending on your needs. The easiest way is to set the backup drives aside, if possible, until you're sure you'll no longer need the data on them, then erase them and start over. Meanwhile start a new backup on one or more empty storage devices. If that solution isn't workable for you, ask for instructions.
*If you don't see any snapshots in Time Machine, exit the time-travel view and then hold down the option key while selecting
          Browse Other Backup Disks...
from the Time Machine menu, which has an icon that looks like a clock running backwards. Select the backups of the computer by its previous name. If you don't have the Time Machine menu, open the Time Machine preference pane in System Preferences and check the box marked
          Show Time Machine in menu bar