Copying the music from the memory stick to the computer is just a copy operation -- you can drag and drop into a folder, for example -- but getting anything onto the iPod so the iPod can play it requires software that can interact with the iPod's internal database. The iPod can't play files that you just copy to it, it's more like installing than copying the files. iTunes is the Apple-approved method for getting music onto an iPod Classic.
An iPod Classic is not "just" a disc drive or storage device. It has a CPU in it and runs software. That software expects the music files to be in its database, and won't play anything that isn't listed there.
Now, if you have replaced the software in the iPod -- Once Upon A Time there was 3rd-party software like Rockbox (
https://www.rockbox.org/) -- or are willing to give up iTunes forever and run a 3rd-party syncing tool, then you don't have to use iTunes. But that is not for the faint-of-heart.