I think there's a kind of 'loading' period where your body absorbs caffeine and normalizes into a caffeinated state. I don't know if the sleep cycle has much to do with it though. In my case, I sleep much better after I have caffeine. I've been sleeping poorly for several months now because of my decision to cut it out.
I might be abnormal (people I live and work with certainly think so!), but I basically never feel 'tired.' I go from feeling 100% to "I need to sleep." This has remained true even after I cut out caffeine, so I can't really relate to how you describe working while tired.
Definitely not a healthy habit, but I used to take an energy drink to bed at night because without the caffeine it would take forever to fall asleep. I'd believed it was an addiction, except now I have been 'clean' of caffeine for so long (without any urges to consume it) and still suffer the same difficulty of falling asleep.
I might be abnormal (people I live and work with certainly think so!), but I basically never feel 'tired.' I go from feeling 100% to "I need to sleep." This has remained true even after I cut out caffeine, so I can't really relate to how you describe working while tired.