In canon, the hat is considered a relic of Gryffindor specifically, which is why his sword can be drawn from it. The Wikia seems pretty clear on this
Quoth the wiki:
The Sorting Hat was sewn roughly one thousand years ago and was merely a normal hat belonging to Godric Gryffindor. When Gryffindor, along with Salazar Slytherin, Rowena Ravenclaw and Helga Hufflepuff, wondered how they would continue to sort the students when the four were dead, Gryffindor pulled his hat off of his head and, with the other founders, enchanted it with brains and some amount of personality. [...] Godric Gryffindor’s Sword, one of the founder’s only other known relics, can be magically pulled out of the hat by any Gryffindor considered worthy, no matter how secure the sword’s location.
It was a joint effort from the beginning.
Also, do you think that the crystal rod thingy is actually Gryffindor’s Crystal Rod Thingy?
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.
Quoth the wiki:
It was a joint effort from the beginning.
Also, do you think that the crystal rod thingy is actually Gryffindor’s Crystal Rod Thingy?
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.
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.