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.