I've not had time to look into the proposals for a while but last I heard there were two key stumbling blocks that could prevent an EU commercial outfit or EU national private pilot from operating third-country aircraft under third-country licences in EU airspace.
First was the gotcha that third-country aircraft cannot be resident in the EU for more than three months at a time. This allows third-country airlines, commercial operators and private pilots to visit, but prevents EU residents from operating under anything other than EASA licences in EASA registered aircraft unless they intend to depart the EU every three months.
Second, in order for EASA to accept third country licences for EU residents, a bi-lateral agreement must be reached between EASA and the country in question, which in the case of an EASA/FAA bi-lateral is as likely as EASA winning an award for 'light-touch regulator of the year'.
I believe a Notice of Proposed Amendment (NPA) dealing specifically with third-country aircraft and licence privileges is being published later this year.
Caveat - all the above could have moved on since I last caught up with the mysterious goings on at EASA HQ, but I suspect the last thing EASA wants to see are EU residents flouting EASA regulations by opting to operate under a different Authority's banner in their airspace.
Regards, jez