Isn't the crucial part here if this is a treaty between the EEC/EC/EU as such and the EFTA states or between the states in the EEC/EC/EU and the EFTA states?

Since the composition of the EEC/EC/EU has changed since, the new member states relation shows if being a EEC/EC/EU member automatically gives EEA memberships.

AGREEMENT ON THE - EEAagreement.pdf

Article 128
Any  European  State  becoming  a  member  of  the  Community  shall,  and  the  Swiss  Confederation  or  any  European  State becoming a member of EFTA may, apply to become a party to this Agreement.  It shall address its application to the EEA Council.

So it isn't automatic, but it is mandatory. The EEA Council could in theory deny membership to new EU members, showing that it is not a package deal even though they go hand in hand.

I think that is a strong argument that the EEC/EC/EU states are members in EEA as such, just as the EFTA states. So it coud be argued that leaving the EU isn't the same as leaving the EEA.

As to why the UK government would want it, I think there are two different answers:

1. It gives a halfway house from where they can conduct further negotiations on their own timetable. A major problem with leaving is the timetable and lack of clear relations to the rest of the world upon leaving.
   
2. The UK government's position on not wanting free movement but wanting access for finance can not be squared, so in the end they might go for something else. For example the EEA.