No, early risers also go to bed when they're tired and wake up when they're done sleeping.
Having to get up at a time that doesn't suit you due to external constraints is something that can affect anyone.
It's just that early risers get tired earlier (and fast) and then often wake up quickly, fully awake. My brother has always been like this.
I'm a night owl. I get tired more slowly, go to bed later, and take a while to fully wake up too.
It seems to me there is a difference in metabolism as well as the time we get tired. Early birds are like sprinters, fast but short distance. Night owls are more like marathon runners, not so fast but keep going longer. I don't have scientific evidence for this, it's simply how it appears to me.