Surely, the whole point of having a 'Display Name' element in a picklist is for it to be used in a selection list to the user. Why is there no ability and the default 'Name' element is used instead. This makes no sense.
We are happy to say that this feature is now available in the updated Common Data Service. You can now use the display name within your apps for Option Sets.
To use this feature you need to be on the latest version of the Common Data Service for Apps. For information about this update in geneneral, see : https://powerapps.microsoft.com/en-us/blog/cds-for-apps-march/
Let me refer you to the other thread on this topic CDS Version / Upgrade.