It was a joint effort from the beginning.

Enchanting the hat was done by all four? Yes, I already tacitly agreed with you about that. But as you can see from your quotation, the Hat is still considered a relic of Godric's specifically. It is aligned with him. It is narratively imperfect for the Hat to serve individual tasks of other founders. On the other hand, Eliezer has already used it to serve a task of Salazar's, so it could do one for Rowena as well. That just doesn't fit as well. Nothing deeper than that.

Also, do you think that the crystal rod thingy is actually Gryffindor's Crystal Rod Thingy?

More likely his than anyone else's. But who knows, maybe it's Helga Hufflepuff's Crystal Rod Thingy, and then all four of them can have done something extra to the hat.