I got a reply from the visa4uk support team. They say appointments will be available later in the month. This answer implies the issue doesn't gave anything to do with my personal situation and application, however, the support stated again that they don't consider such things in their replies.
EDIT: I have since got a very short-term appointment. When I went to the visa center, there were a lot of people desperate for appointments because their travel date was really soon. The visa center staff explained that they were very busy, hence the difficulties with getting an apointment online (later months seem to be made available only with the passage of time). The short-term appointments like the one I got (made available Mon, 17.12 for Wed, 19.12) were, and I quote the staff, "a fluke, we agreed to do one extra day out of the goodness of our hearts".
The staff said to one very desperate lady that they'd contact UKVI and see if they can do anything. I left by the time they tried so I don't know whether they were successful, but the staff didn't seem optimistic.
However, for really urgent cases it seems indeed advisable to contact UKVI (via visa4uk support) because they make the appointments available and not the visa center staff.