Ancient DNA Reveals the First Known Neanderthal Family

[GHurier]

The Spy Neanderthal paper says that Spy and Mezmaiskaya share a Y clade that falls outside of modern human variation, including A00 in the modern human clade. So why do you figure Spy has A0, TanTin?

According to Petr et al. 2020 (https://www.science.org/doi/10.1126/science.abb6460), Spy is indeed outside of human variations.
The pre-A00 node for Modern Humans and late-Neanderthals is not even doubteous and is found for the 100 bootstraps replications.

The TMRCA they obtained with Modern humans is ~370 kyr ago (way before A00 splitting).

A claim that late-Neanderthals are under modern TMRCA would need a proper analysis with a peer reviewed publication.
I haven't seen anything convincing on this topic so far.

So then wouldn't it be more accurate to say that modern human y-DNA is actually Neanderthal derived instead of the other way around?

There is a chance it is the case (but we lack early samples to make strong claims).
I might put a dedicated post about that, that would discuss the mtDNA and Y-chr splitting timings and the possible implications.