Of course, I have always used the same account, so the game is linked to my profile. 
336 days ago(+3)
Indeed, this is a very odd issue.
Because per itch.io rules, download key owners also have access to the external key (Steam Key), they stated clearly here: https://itch.io/docs/creators/download-keys
You claimed the game from "The Arrival of Nightwind" promotion, it has tied to your account and given you permanent access, as well as giving you the "Key Owner" status, you should be able to access the external key.
I checked that I have supplied enough Steam Keys, or else there will be a "There are no keys available at this time, try again later" message on the download page instead of showing nothing.
Nevertheless, I already contacted itch.io support days ago, hopefully, I can get a reply soon and solve this issue.
336 days ago(+2)
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!
334 days ago(+1)
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! 😉