I think it ultimately came from the Caucasus region, into Anatolian, not the Levant into Anatolia. If so, it probably came by way of the Bronze Age pulse of Iran_N-like ancestry.
Part of my reasoning:
Only about 5% of LBA Anatolian autosomal DNA is attributed to Levantine Farmers, while over 1/3rd is attributed to Iran/Caucasus.
I don't understand the relevance of autosomal DNA here. This is why I find your argument flawed:
There are a handful of Greek J-L70 lines identified currently. For this, let's assume they all arrived in Greece via an Anatolia route (I don't believe all the Greek L70 necessarily passed through Anatolia). After leaving the Levant for Anatolia (some time between 1600 BCE and 1300 BCE), these men would ultimately assimilate amongst the Anatolians, so that by the time they would have arrived in Greece during the LBA, they would be entirely LBA Anatolian in their autosomal ancestry, but their Y's would still be J-L70. It doesn't stop there, they would have continued mixing in Greece, ultimately becoming autosomally modern Greek today, with their lines remaining J-L70. Obviously when their ancestors would have arrived from the Levant, it is unlikely that they would remain Levantine - point being people are bound to mix and mingle, with the "original" autosomal ancestry dying out. [An exception would be the Jewish populations, who have always retained a large proportion of "original" ancestry]
Additionally, by the time J-L70 men from the Levant arrived in Anatolia, they would not have been 100% Levantine farmer. They would have ancestry from several sources, being Natufian, Anatolian Farmer, Iranian Farmer, CHG, even some Steppe. So one can easily say some of the non-Levantine Farmer ancestry in BA Anatolians could have been from them. But ultimately I have no interest in debating on whether Italians and Greeks have Levantine ancestry (although given the presence of J-L70 among them - not to mention other Levantine uniparentals in these regions, we can basically confirm they had Levantine ancestors at some point).
It is not clear on what basis you think J-L70 came from the Caucasus - is it just due to autosomal ancestry? In which case, you'd make a stronger argument by saying it just originated in Anatolia.
You might say that the first point I raised can easily be flipped around, with Caucasian men arriving in the Levant, assimilating, and being responsible for the J-L70 in the Levant. But we can dismiss this for several reasons (which doubles as evidence J-L70 never came from the Caucasus):
1) J-L70 is a rare clade in the Caucasus, except amongst Jews (Mizrahi - Juhurim) and Armenians, however Herrera et al, (2011) concluded that most of the paternal lineages studied amongst Armenians were introduced from the Levant. In this specific paper, they found J-M318 in 5% of Armenians from Lake Van - J-M318 is Jewish in origin, reinforcing that at least some of Armenian J-L70 is Levantine. Several other Armenian lineages fall under J-Z40772, which shares a common ancestor will several Jewish branches at 2700 ybp, and with a large Jewish branch at 2900 ybp, reinforcing once again a Levantine origin. We also need to remember that the Caucasus is a more heavily tested region, particularly than the Levant and Anatolia.
2) Considering the greater number of Caucasian men tested, we find no Basal branches of J-L70 amongst them
3) Considering the greater number of Caucasian men tested, we don't find any significant diversity of J-L70 branches amongst them
4) J-L70's main growth period ends abruptly with the Late Bronze Age collapse, solidifying J-L70 being heavily nested within the Eastern Mediterranean - I don't suppose the Sea People's invaded the Caucasus as well?
I think it was likely spread to Greece via Anatolia, originating from the Caucasus region. The paternal lineage could have been spread also to the Levant from Anatolia, rather than the other way around. We can see by looking at different time periods that there was an Anatolian/Iran_N intrusion into the levant. Just look at Levant_N vs Levant_BA:
Having dismissed coming from the Caucasus, the next thing to address is spreading from Anatolia. While more likely than the Caucasus, there still is not convincing evidence for this. J-L70 is not a Neolithic marker, and could not have spread via Neolithic movements. J-Z387 is a different story, with it likely arriving in the Levant via Late Neolithic Mesopotamians (with the ancestor, J-L24 and J-L25 originating in Iran). I did address in my first post why it's not really possible for J-L70 to have spread from Anatolia, with one set of branches going to Greece, while another went to the Levant.
The phylogeny simply tells a different story. It's not like we see a clear cluster of Greek branches, and then a clear cluster of Levantine branches -
all of J-L70's branches are mixed. This suggests J-L70 diversified amongst a certain culture, or in a certain region before spreading all over. Additionally, we find Jewish branches throughout J-L70, whereas we would need Jewish branches in just one neck of the woods for a spread from Anatolia to work. The presence of Jewish branches throughout the phylogeny strongly supports a spread from the Levant, especially since Jews tended to remain insular.
I'll just repost why a Levantine origin for J-L70 is most likely, and has the greatest evidence:
"A Levantine origin for J-L70 makes the most sense given the great branch diversity, higher frequencies than other regions, and presence of several layers of Basal branches (J-L70* in Jordan, J-FT340863 in an Sinai Egyptian and Palestinian, J-Z387* in several Sephardic Jews, and two different J-FGC35503* branches in Lebanon). Should J-L70 not originate in the Levant, this is all very coincidental. The nail on the hammer for a Levantine origin for J-L70 are the 15 distinct, distant Jewish branches. No similarly aged clade has as many distinct Jewish lineages - save for perhaps J-Y2919, but that's 600 years older."
So pretty much, this is the likely migration path we're dealing with: J-L25 (Iran) >...> J-Z387 (Mesopotamia) > J-L70 (Levant)
