The "separate" farming groups of the Near East didn't stay separate for long. The first Anatolian farmers who came to Europe already had "Natufian" like ancestry, and the beginnings of Zagros ancestry too.
Succeeding waves carried more Caucasus like ancestry.
There's also the fact that the Greek Neolithic was a bit different.
In the Near East, everyone mixed. Caucasus like ancestry moved south and west, and Anatolian like ancestry moved north and east. Yes, there's a cline, with Caucasus like ancestry increasing as you go north, but Anatolian like ancestry made it all the way to the Armenians.
As for places like Iraq, there were large tribal movements into it in the Muslim period, and into Jordan, so things changed again.
If you're getting this stronger "Levantine" signal based on amateur analysis I would be very wary. The academics have analyzed them, and personally I would stick with that.
There is either no or very little Natufian content in the vast majority of early European samples.
What have the academics said about the Levantine signals attached to these particular samples? Have they analysed the Levantine content and come up with a different answer? If they have, I would be interested to see it. If they have not, then I see no good reason to dispute it, particularly as one of the samples has a Middle Eastern yDNA lineage.