I’m not a fan of battle style but is it really fundamentally different? Eddie’s patent is more like rotating the deck about the z axis and holding it like a guitar. Which would be pretty cool and ridiculous come to think of it :-)
That, and the footprint of a setup is reduced when decks are placed battle style. This is important because turntablism requires a lot of very quick manipulation of both decks and mixer, and it's easier to do this if everything is a lot more compact.
That's also why you typically see turntablists using a simple, small two-channel mixer while other DJs tend to employ wider, bulkier 4+-channel mixers.
To OP's point I could've maybe dropped the word "fundamentally" in my post because at the end of the day the fundamentals remain the same, as do the fundamentals of the guitar in the patent vs regular use.