There is some logic here, but it's tricky and thin.
If information is released publicly, there's still reasons to keep it classified. Once it's unclassified, anybody with a FOIA request can verify it, but if it's still classified there's at least some plausible deniability. The government can claim that some of the documents don't really mean what they say, or were faked, or whatever. It may not apply in any particular situation, but somebody has to make that determination, and until formally declassified it's still classified.
It's easier and simpler to tell people not to read classified information than to tell them not to read classified information except from publicly available sources. It makes investigation easier: if you know that document X should not be in the browser cache, it's easier if you don't have to figure out where it came from.
Security clearances are not mandated except if you want a certain job, but are rather voluntary, and in this case there's some reason to have strict rules that are clear and don't involve judgment.