You're confusing two different parties involved in the monitoring here: the FCC and the station owner.

The station owner likely knew within a few hours to a few days that the tower went offline. At the very least the power bill would have been very low or non-existent that month which would make it pretty obvious. And to your point they could tune their FM radio to their station frequency and see if it works.

The FCC is a federal organization. I doubt they have an office in Jasper Alabama. So the FCC monitoring is the one in question here. Because as soon as the FCC notices the AM station is down, then the FM translation (which is what the station was really marketed around and gaining its viewership from) is shut down. Which means the station owner is out of a job. The FCC's monitoring is probably not realtime and I could see someone at the FCC getting an alert that this AM station in Jasper Alabama is down and they do think to themselves the monitoring is bad or flaky and don't immediately panic. The FCC isn't going to be able to just tune their radio to the AM station to check since they aren't in range of the station.