As far as EEF is concerned, 20% WHG is the
upper limit; Lazaridis stated that it could be anywhere from a few percent
up to 20%. Until we have some samples from the Near East we won't know.
In my opinion, the results of this very sophisticated study are becoming increasingly misunderstood as they are simplified, and perhaps even slightly distorted to make agenda driven points. It's like that kid's game, where a statement is made, then passed along a chain, and by the time you get to the end, the message is totally garbled, only worse, because some of the players are deliberately twisting the words.
I would concur with Sile's point about EEF being "Middle Eastern". EEF is EEF, a very particular mix that existed at a very particular time. I think it can be most accurately described as the signature of the early European farmers. I don't know how many times certain things need to be repeated before they sink in. The people who came to Europe bringing agriculture with them came from the region of the greater Near East, some perhaps from the Levant, but many from the more northern Near East. Regardless, those people are not identical to the people who currently inhabit the Near East, who now have a large ANE proportion, as well as an additional slice of SSA. They arrived 8-9,000 years ago.
Until we get samples tested from the Italian and southeastern European, including the Aegean, areas, we won't know whether the hunter gatherers in that area were like the hunter gatherers of central and northern Europe, or if perhaps they were not that different from the incoming farmers, meaning that the signature might indeed be Mesolithic and therefore older in Europe than 8,000 years ago, although there still might have been a substantial movement of actual people from, say, the Levant. The distance between Anatolia and the Balkans is the width of the Hellespont, which is only about 1.2 kilometers wide. Could the people have been so very different? Perhaps it was an insurmountable barrier in the Mesolithic. I don't know. I'm not willing to go out on a limb and say they weren't that different, but I think the results coming from the Bean project about the Mesolithic hunter gatherers in Greece having no mtDNA "U", and instead the Mesolithic and Neolithic samples having very small FST distances, and carrying mtDNA signatures X,K,J,H and T, is very suggestive.
See:
http://dienekes.blogspot.com/2012/12/talk-by-christina-papageorgopoulou-on.html
Furthermore, if you go back in time, the northern and central European hunter-gatherers arrived in Europe either through the Middle East or Asia. Europe is a destination, not a source.
Finally, I don't know how many times it needs to be said...hunter-gatherers don't form an "ethnic" or even geographic group.
All humans were hunter-gatherers until they adopted agriculture... some just did it earlier.