Does that mean if a student is eligible they 'always' do it? In other words is it always done, or are there other determining factors on why a kid would not reclassify if the option is there?
No, they do not always do it - generally, there's additional course work to be taken and some kids just prefer to stay where they are and hit college as a more mature player/person.
It's fairly common for athletes to stay back a year early on - the extra year allows them mature more and presumably gain more interest from colleges down the road. (I think that's rampant in some sports like lacrosse.) In basketball, you frequently have an inner city kid who might not be attending the best school and when that kid's talent become apparent, they'll transfer to a better school or a prep.
Take Shabazz Napier.
He attended Charlestown in Boston Public system. Not the worst school, but low college-readiness scores, etc.
After his sophomore year, he transferred to Lawrence Prep. They evaluated his transcript and enrolled him as a sophomore. When UConn offered in what was his junior year, he already had nearly four years of credits accumulated and it was just a matter of taking additional coursework in the summer and successfully passing the standardized tests.
Now, if he had been reclassified as a sophomore and none of his previous sophomore year work was acceptable, he would have had no chance at reclassifying.
Andre Drummond was a different case - he was completely eligible as a high school senior, but his mother wanted him to prep for a year before college. In that instance, his recruiting "class" was an artificial distinction. He didn't reclassify, he just changed his mind.