First of all, when the iPod is connected to your iTunes Library, check on the iPod/Summary pane to make sure that Sync only ticked songs and videos has been selected. If not, select it now.
There's a possibility that a Sync may have finished before you select the above option, so press the Sync button after making a change like that.
Well, if you have Sync only ticked songs and videos turned on, it has to work, that's the point of it.
It wouldn't work if a Sync had not actually happened. If you see indications that the Sync is proceeding, then it should work.
Having said that, I admit that sometimes I have to press the Sync button second time in order for my iPod Touch to Sync fully. In other words, some parts of the Sync take place, while others do not. But - note that this is on my iPod Touch, which is a far more complex device than the Classic. I have never had this happen on my Classic and I use both iPods a lot, each with it's own Library.
The only thing I can suggest is put the check-marks back on, connect the iPod to iTunes and wait for the Sync to complete and then, without disconnecting the iPod, turn the check marks back off again and press Sync again. Once both Syncs have taken place, disconnect the iPod and check it.
If that sorts the issue out, then I'd say it was a glitch (not a bug). (A glitch is "just one of those things that happen once in a Blue Moon".) If that doesn't sort it, then I really don't know what else to suggest, apart from a Restore.
Restoring the iPod (using the Restore button in iTunes) will remove all the content from the iPod and then put back onto the iPod, only what it finds in the Library that is set to be put on the iPod. In other words, starting again.
It's a drastic step, because it takes a while. Note also, that everything is deleted from the iPod during the Restore process. iTunes can then only put onto the iPod what's in your Library.