IIRC it takes about 30ish minutes for caffeine to "kick in". So if you have a bit, then take a nap, it can give you a nice 20ish minute power nap, then naturally wake you up so you don't feel groggy. The key is to be able to fall asleep quickly enough to have a decent power nap before it kicks in.
Different organisms react differently to the same chemical substances. There's a study linking ADHD and paradoxical reaction to stimulants:
Many people with ADHD have next to no reaction or react paradoxically to caffeine (coffee/black tea/cola) and other stimulants (nicotine, “Red Bull,” amphetamines).
As I'm writing this reply, I have not yet been able to find any other studies that delve deeper into the matter, but it seems like it's indeed a known phenomena.
My son is ADHD and autistic and caffeine calms him down. Sometimes when he misses his meds we will give him some coffee or soda to get him through to his next dose.
There's exhaustion, which builds up and is relieved gradually with sleep, and is effectively masked by caffeine. Then there's the feeling of incomplete/interrupted sleep; maybe the quality of sleep was poor somehow, maybe you were woken up by an alarm in the middle of a sleep cycle, but it's painful, cognitively debilitating, and sticks around like a splinter in your brain. Caffeine doesn't help it at all, but it can be completely relieved by a nap in which you succeed in falling completely asleep even just for a few minutes, which caffeine does not prevent from happening.
I don't know how much of that is objectively how it works for everyone, but that's how I understand it.
This is purely personal experience, but I've found that the usually warm beverage that caffeine comes in + the increased heart rate helps to warm up my muscles, which makes the muscles relax more easily.