Oops, I guess I wasn't very precise in my wording before. Let me explain it better now. I think the sum of J1, J2 and E1b1b nowadays is too large, in some Iberian regions verging on 1/3, to have left relatively little North African and Levantine/Southwest Asian admixture (that's what I meant by Eastern Mediterranean, sorry I forgot for a moment that Greece and Anatolia is also on the eastern side of the Mediterranean... LOL!). I also think that such a heavy contribution from non-Europeans would've probably shifted Iberians significantly on the PCA charts away from other Western/Southwestern Europeans and much closer to West Asian/North African populations (unless we assume that they were also affected by the waves of North Africans and West Asians... unlikely in my opinion).
There's also the issue that the non-"steppic" Y-DNA haplogroups are found in large percentages in places that were never under the direct and permanent dominion of Arabs, Berbers and Carthaginians (E1b1b IIRC is found in comparatively significant proportions in Galicia for example).
Yes, but that was a long time before J1 and J2 may have arrived in larger percentages in Iberia, and definitely way earlier than the supposed arrival of these haplogroups mainly with colonization processes as late as those of the Carthaginians, Arabs and Berbers. More than enough time for J1 and J2 to be associated with completely different autosomal makeups depending on the region where they came from.
My point is that, since modern Iberians are still firmly on the "European side" and not that close to North Africans or Levantines, much of those J1, J2, E1b1b or G2a may have come from other European populations, already heavily shifted to EEF and, much less so, Steppe BA, during the Chalcolithic and Bronze Age and as late as the Iron Age (Celtiberians - clearly a late arrival on Iberia considering their language was just too close to Gaulish to be indigenous to Iberia since the Bronze Age - and Romans).
Considering the possibility that these haplogroups arrived much earlier, during the Chalcolithic or Bronze Age, more or less contemporary to the "steppicization" of Iberia, I think it's more likely that much of it came from heavily EEF/ANF+CHG regions like Bronze Age Anatolia/Greece, so that their contribution would be masked in Iberia due to the heavy presence of Neolithic Iberian EEF and BA Bell Beaker-like CHG/EHG + EEF. If they had mostly come from the Levant or North Africa, wouldn't we expect much more Natufian or Levant_Neolithic in modern Iberians than there is in fact?
I also have a hard time believing that Carthaginians left a more widespread and long-lasting genetic impact than the Romans, who had in Iberia some of the highest concentration of Roman colonies, controlled virtually the entire peninsula and didn't actually get beaten by invaders until more than 500 years after their conquest.
But last of all, I'd really want to know if the J1, J2 and E1b1b subclades that are most frequently found in Iberia are really closer to those found in North Africa, Levant and Arabia (consistent with the idea that the Carthaginians, Arabs and Berbers really contributed much more than we'd thought to the Iberian genetic pool) - and not to those found in the rest of Europe.