Usually O'Reilly is not Live. Usually recorded in late afternoon.
Correct, O'Reilly is probably taped at around 4 or 5 PM Eastern time.
There are many instances where breaking news has occured between 5 and 8 PM Eastern time and O'Reilly's show doesn't mention that breaking news.
It's probably in his contract that the show is taped in the late afternoon so he can have a 9-5 job.