By the way, have you connected your itch.io account with your Steam account?
It says "A Steam account is necessary to claim Steam keys from your purchases where available." So it might be worth a try.
I claimed the game in the same giveaway and had already made an itch.io account prior to that, and had linked my Steam account with it, and I do not see a button for the Steam key. I've heard itch has issues with Steam keys in general, especially with pages that didn't have Steam keys before...good luck with the support, anyway!
The truth is that I was unaware of this feature, but I just linked it and it hasn't changed anything, however, I don't think it's fair to give you so much work, just for a key, for my part, I don't find a real difference between playing it here or on Steam. Although I will of course report back if this situation is resolved, best regards! 😉