No.. My problem is that there is no "claim steam key" section and no "get steam key" button in the download page
It looks like this is out of my control. There are enough keys and they are claimable. If this issue persists on other games too, I guess the itch.io support can help you with that. Let me know what they say about the issue.
Yes I think it might be the website's problem, but thanks for helping me out!