Here here! It is long due.
They can keep the animations because they are an important part of the game, but there is no actual need to actually lock your skills during animation.
Let the animation be canceled by the next input.
Of if you don't want triple weaves to happen, just reduce the animation locks enough to let double weaves be doable by people with 150+ ping.
World of Warcraft does it and that never had a complaint from the community, nor has it broken the game.
And yes, I believe FFXIV is a much better game (in my oppinion, of course), but the animation locks are a major downside for this game and make people living one country (or more) away from the server feel like they can't play the same game that "natives" do.