Legend of Korra

That doesn't sound promising, I would have liked if the White and Red Lotus clashed head to head.

Would be pretty one-sided given how incompetent the White Lotus has become.

The Avatar wasn't in the original series, Iroh was. When Iroh passed away, Aang was probably too busy with Republic City and the rest of the world to be able to devote his time wholly to the White Lotus, and so someone else most likely took over. Even if he was, when he died there's no way they would have given the position to Korra immediately, especially before she was a fully realized Avatar, so if she were going to become the leader of the White Lotus we would have seen that happen.

I was thinking in a more de facto sense and less de jure. Yes there is an official leader of the White Lotus and the White Lotus operates independently unless the Avatar has personal requests. I mean, it's quite clear that the White Lotus has devolved into a quasi-army for the Avatar (no matter how incompetent it may be).