I wouldn't use J2 as an argument, because just like H and E1b, it seems to have been common in various early farmer groups, which were otherwise not much more Eastern shifted than the LBK farmers, sometimes even more WHG actually. Because of this, we need to reconstruct J2 and its subclades migrations first, before using it for any sort of argument in favour of an earlier Iranian ancestry spread.
Since you mentioned Minoans: Its so low in those, that even if they would have replaced local, earlier EEF groups completely, it wouldn't suffice. The gap is just too big without the later, Hellenistic and Roman era gene flow. At the same time, however, I think you are right that there was a constant trickling in too. A lot of samples hint at this kind of slow effect, just like the Northern shift in Italy didn't end with the migration period, but continued with slow trickling in from Central Europe directly and indirectly from Northern Italy to Southern Italy.
By the way, I wondered about the Molise sampels and their special case, but I guess some of it can be explained by the Slavic settlement in the region. Its interesting to compre Molise with Marche. There are just 2 samples form Molise, but these have above average Slavic admixture, while the Marche samples have one with higher Slavic, but the rest being more Celto-Germanic shifted in comparison. Quite interesting, but probably the result of:
https://en.wikipedia.org/wiki/Molise_Croats
Does anyone know results of "full blooded" Molise Slavs? If such even exist up to this point.