I got some answers when I swung by the hotel on my lunch break. The hotel is indeed open 365. I asked if that was always the case, explaining that in the first week or so of reservations, I wasn't able to book on days the park was closed. I couldn't get an answer on that one.
Maybe it was a website glitch (there were many in the early days of this resort) or those days simply hadn't loaded yet.
What's interesting is that you can book those rooms without knowing that the park is unavailable during your stay. There is no language on those pages alerting guests to that fact.
Anyway, thanks for keeping me honest!