Annex III covers the basic requirements for pilots licensing under the EASA regime, but the bit of Article 7 I referred to excludes Annex III when there is reliance on licences issued by a third country, otherwise the 7th sub paragraph makes no sense.
Different Annex IIIs and different Art 7s! The Essential Requirements for Pilot Licensing are Annex III of the Basic Regulation.
Art 7 of the FCL regulations refers to Annex III to the FCL regulations. The latter pair are the "detailed regulations" you refer to in your post above.
I hope you're right and I'm wrong.