getting kicked from end game pfs as soon as I join?

[whiskeybravo]

Furthermore, you have no ownership over the battle log information nor your character's names—both of which being what FFlogs users.

Probably one of the most important points.

All aspects of the Game (including but not limited to the Game computer code, updates, patches, expansion packs, characters, character names, character data, virtual items, character profile information, (non-personal) account information, chat transcripts, artwork, plot, dialogue, locations, sounds and music) are owned by Square Enix or its licensors. You do not have any property rights in the service account you use to play the Game ("FINAL FANTASY XIV Service Account" or "FFXIV Service Account").

So, technically it's SE's personal data, not yours.

Not just that. By using this argument, SE would actually have to basically censor everyone's username in every instance, not just battle logs (and by extension parses).

Oh it's better than that. If you follow the claim to it's logical conclusion then it'd be inappropriate to ever even use someone's name in real life. "Hey Person I used to call Mom, how are you today?"