Need to restore from Time Machine but it can't find it!

I did another round with Apple support - as they also could not understand how I could be successfully seeing/using AFP volumes on my ReadyNAS, successfully executing time machine backups to another volume -- but not being able to access those backups using any of the Apple "restore" applicactions (Recovery or Migration Assistant).  At their suggestion I took a separate machine (I used an older MacAir) and installed the High Sierra beta, to see if that version of Migration Assistant or Recovery had the same issue with not allowing authentication to the ReadyNAS backups.  It exhibited exactly the same behavior that you described in your post (default to 'root' userid, not accepting credentials that were configured into the Time Machine service on the NAS, either directly in that dialog box or via a fully-qualified path using the 'other server' option in the restore utilities.  

Very frustrating that I have not been able to get Apple to tell me what protocol I need to get a NAS to support in order to have restore from Time Machine work - nor have I been able to get anyone from NetGear to even acknowledge that this is a problem, or tell me what other model/upgrade might be required to get this back functioning the way it had been working for years before.