The only way lots of H could be from Mesolithic Europeans IMO is if SE Europe was autosomally not WHG and had lots of H. IMO, there isn't enough data on H to have an idea what its history is. We need full sequences Hs from all over West Eurasia. We don't currently have that.
If it weren't for ancient mtDNA I'd think J1c and T2b colonized Europe after the Ice age, because there's not much of it in West Asia. But, ancient DNA has shown modern people aren't representative of ancient people in the same region. So, what may appear to be European H(or J or T) today is may not be from Europe.
The way I see it is that J1c and T2b were already present in the Balkans AND Anatolia when the first farmers arrived from the Fertile Crescent. T2b was found in Mesolithic Russia and Sweden, so it must have expanded early, probably just after the LGM. It could have originated in the Caucasus or anywhere around the Caspian Sea, then spread north to Russia (then Sweden), and west to Anatolia and the Balkans.
Let's not forget that agriculture arose about 11,500 years ago in the Fertile Crescent, but these early farmers remained in the Fertile Crescent for some 2500 years before they decided to expand to northern and western Anatolia and Europe. Once they had started it only took them a few centuries to reach Germany and France from Greece.
The original farmers (9500-6500 BCE) were Basal Eurasian in term of autosomal DNA, and also carried Basal Eurasian Y-DNA (G2a and surely also G2b in the south) and mtDNA (N1, N2, W, X). We have seen times and again that Paleolithic and Mesolithic tribes were fairly homogeneous in terms of both admixture and haplogroups. By this logic, the first farmers in the Fertile Crescent must have carried overwhelmingly Basal Eurasian haplogroups. The only notable exceptions are H5 and K1a. But who knows, they could have been absorbed from close neighours within or just outside the Fertile Crescent between 9500 and 6500 BCE. Since R1b was already in the region at the time, domesticating cattle in the northern Fertile Crescent, it is not unreasonable that G2a farmers picked up lineages like H5 or even U8b1 (the ancestor of K*) and K1a from intermarriages with their cattle herding neighbours. After all H is overwhelmingly European and U8 has been found in Paleolithic and Mesolithic Europe, including Russia, where other R1 tribes lived.
In my eyes, the J1c, T2b, U3 found in the Bacin site were all assimilated Mesolithic Anatolians. I have
linked mtDNA U3 to Y-haplogroup J1 and T1, which both seem to have expanded from the Caucasus region. J1c was surely found in the Balkans and Anatolia, while T2b would have been all over the Black Sea and Caspian Sea periphery. In other words, T2b and U3 were almost surely found among J1, J2 and T1 tribes living at the northern edge of the Fertile Crescent, while J1c could have belonged to a yet undefined Mesolithic Aegean tribe (C1a2, I2c, H2 ?), with other mt-haplogroups like various H subclades.