Update to this story.
Basically I completely failed to get Zotac to even ask me to send them my graphics card anywhere. The US branch keeps absolving themselves of the responsibility. The responsible region/branch, based on the Zotac official website (Eurotech, see the screenshot), just doesn't answer emails and are totally rude.
No phone call attempts or emails helped. Everyone just keeps throwing the responsibility onto someone else.
I think I'm gonna fix it myself, I bought a spare socket, and I'm never, ever, ever buying Zotac ever again.
Let me know if you have any advice/questions. Otherwise, thank you for all the advice.