Or do they just read the threads that seem ripe for abuse and only check in on others when posts are reported?
In addition to reading/participating in threads they are actually interested in of course.
It just seems like read every thread for modding would be more than a full time job. As well as being extremely tedious.
One of the reasons I could never be a professional proofreader/editor. You have to read the bad along with the good.
I get the impression that most moderation is done in response to reports from other posters. It seems to me there have been threads in which a mod participated but let some infraction slide because they failed to notice it or just apparently thought it was not a great big damn deal.
For them to proactively moderate every thread, reading every post, would be too much to ask for persons of their pay grade.
I have no life, so I read a lot of the threads. I got nothing much better to do. But I miss a bunch of shit. If they concentrate specifically on the forums they mod, I don’t see how they would miss much. Ain’t much to mod on in the thread games or cafe, those should be easier. The pit and others would be more challenging, I think. IMO.
We do not ready every thread and every post. However, we are expected to participate in the forums that we moderate, so we do read a lot of them. To that end, we are generally assigned to forums that we like to participate in.
We do not solely moderate based on post reports. Since we do read and participate in threads in the forums that we moderate, we will often see something and will moderate it before a post report comes in. We do very much appreciate post reports though, and often something gets reported that we did not see.
The moderation staff here is 100 percent volunteers. We are not paid. We do not have fixed schedules. We try to have enough moderators per forum so that we can quickly respond to most things, but sometimes our schedules line up in such a way that things get missed and sometimes we end up with poor coverage on a temporary basis due to someone being on vacation or otherwise being unavailable. We will often temporarily cover other forums in such cases.
We generally do not have the time to respond personally to every post report. As I said though, we very much appreciate those post reports, so if you see something that you think needs our attention, please report it. Even if you aren’t sure, just go ahead and report it. We get a lot of reports that start out with something like “I’m not sure if this is allowed or not…” We would much rather have too many post reports to sort through than too few.
I try to read as many threads as I can. My order of priority when I get on the boards is to read all thread reports. Then I try to read and catch up on any thread that I’ve participated in. Then I go through and read threads that seem interesting. Depending on my work and home schedule, there were times that meant I read nearly 100% of threads in the forums I mod. Lately I’ve had a lot of work and extra curricular activity so that means I’ve read slightly less.
So in a sense you are priceless and/or protecting your amateur status? 
I’d say that, yes, post reports are the first line of defense on dealing with problem posts and threads. Squeaky wheel and all that sort of thing.
But we try to read threads. And we are certainly aware in GD and Elections which threads - and which posters - are most likely to cause trouble and create headaches so those threads may get a bit more attention.
In short, no, every thread is not read. But we try to read enough to know where things will get weird.