Hattic, PIE Myth

I was looking around for dirt on Hattians in relation to the wanax
thing and came across some stuff that was kind of interesting,

For one thing Hattic Taru is clearly related to Hittite Tarhunt,
which means we got an IE thunder god recalling Norse Thor, Celtic
Taranis, Slavic Perun in Hattic land.

Also there is Hattic Papaya, a goddess of fate, death. Compare to
Baba Yaga. Sun god Tiwaz, resembling Tyr or Old English Tiw. Hattic
Kurunta, similar again in name and function to Cernunnos et al.

It all says to me that a lot of old IE myth was very similar to Old
Hattic myths. That raises some interesting questions about how
exactly these cultures share the same myths.

The traditional explanation I've read is that the Hittites adopted
these deities from the Hattians when they arrive. So, how did the
Celts then adopt Taranis? I thought traditional models have Anatolian
IE breaking out of contact with other branches at a rather early
date .

Did the Hattic dieties diffuse across IE Eurasia at a post-Hittite
date? Or are these similarities pre-existing, running against
traditional views, an ancient mythos in shared by the Hattians and IE
alike. Or could Hattian cultural influence have gradually permeated
IE previous to the actual Hittite heyday? I am curious what others
think.

-Mr. Caws