Malsori, you are mixing apples with oranges. First, because Southwest Asian and Southern are not the same, and you assume that its huge presence in the Middle East makes both things equivalent. Wrong.
If you check the Fst distances, you'll see that the Southern component is a lot more removed from Black African clusters than the Southwest Asian is. Sardinians and Basques (as well as some Iberians), show very little West Asian and Southwest Asian, which are no way absent among Middle Eastern populations. If the resolution is very high, those populations show very high percents of Southwest Asian, and if the K's are lower, they get substantial West Asian despite being modal in Southern. So it's not clear at all that Southern must have come with Neolithic farmers, it could have been in Europe even long before considering its non despreciable presence in Southern Europe.
Another explanation is that Neolithic Farmers were not as Middle Eastern as some people thinks, exactly the same valid IMO. Simply because, as I said in other threads, what is identified nowadays as Middle Eastern, is a composite of West Asian and hints of African superposed to the Mediterranean substratum. Hence, quite different considering what Southern in its "pure" form is: very distant from inner African groups, and not specially close to the West Asian cluster according to K7b.
What I say it's more evident in the K10a run, check what Dienekes' said about Mediterranean (Modal in Sardinians, and quite high in both Basques and Iberians):
As for the African/Sub-Saharan components, they tend to be closer to the Southwest Asian/Red Sea components, not the Mediterranean/Atlantic_Med one.
The Mediterranean components appear to be the most remote ones overall (also evidenced by the fact that Basques and Sardinians nearly always form the peak in the West/East Eurasian/African triangle), which makes sense since the region where the Mediterranean/Atlantic_Med component is modal is most remote from both Africa and Asia along the land migration routes.