It's open 7 days a week, because it is also used by MBTA - who have their own employees. (The same is true at BOS, BBY and BON.) Amtrak has only 1 daily train to/from WOR.
I looked up the Lake Shore Limited, which runs 7 days a week. It's rather odd that they have a ticket window but no QT.
Actually it's not odd at all. QT's are used at the most heavily used Amtrak Stations and stations that see multiple trains a day. WOR sees two Amtrak trains a day. Stations on the NEC see plenty of trains. Every station in the NEC has a QT.
However, Worcester has a ticket window and baggage service. I've never heard of a ticket window combined with no QT kiosk. If anything, I would have expected a QT kiosk only and a no ticket window, so that ther would be some way to get the ticket on a weekend.
I mentioned Reno, which sees two trains a day plus a few buses. Another one I've been to is Roseville, CA. It sees only a few trains per day, has no ticket window, and has a QT kiosk. They do have bus service though.