I doubt it will really enhance competition. Until now, barely any EU airlines are using the right to fly within the entire EU. Notable exceptions are Ryanair and Easyjet, there still is no LH flight from CDG to FCO or AF flight from FRA to HAM or wherever. Only the number of codeshares has increased, but only within the alliances.
So whoever wants to flyto the US with LH from CDG will still have to go via FRA, as it simply is LHs main hub. Chances for nonstop LH or BA flights the US (or other destinations) are very very slim, as they lack connecting traffic at these airports. Also the new rules might actually lead to higher prices, as smaller airlines will now be merged into one of the large three, leading to less competition.
Also, who says other contries will accept the EU as contract party, as it doesn't possess any real souverign status? What if, Japan, China, or whoever says they won't sign a contrct with the EU, only with the individual states, as the EU is not a "country" in classic terms?