I think it was implied that her Trump card NP “Peidra de sol” on the top of her temple in Babylonia was giving her that authority and not the Temple itself. Considering all the participants are summoned with their NP’s, she should have it.
Even excluding that.
Xiuhcoatl: Flame, Burn the Gods to Ashes (炎、神をも焼き尽くせ, Honō, Kami wo mo Yaki Tsukuse ?) is the Noble Phantasm of Quetzalcoatl. The second coming of the flames that burned down her own temple in order to prevent the Evil God Tezcatlipoca from acquiring its many treasures, during a time when Quetzalcoatl was leaving the Aztecs. Flames envelop the surroundings, and enemies are temporarily prevented from invoking the True Name release of their Noble Phantasms.
I believe this might be useful to not let her invoke Karna’s Brahmastra or VS.
Quetz could keep up with Kingu who basically possesses all skills of Enki + a Holy grail within him and his chains couldn’t stop her because of her sheer force (and that’s without her using any NP’s).
Quetz with her NP deployed could vaporize most of the Chaos tide of Tiamat flowing directly from Tiamat and helped in killing her Horn too. Mind you, this is all after her power/Divinity was nerfed by half thanks to Merlin trolling.
Karna is definitely strong too and VS might kill Quetz, but I think Quetz would win this 7 times of 10.