Why would you need to annoy someone, when you can incite people to take specific actions? Strife causes unpredictable outcomes and isn't something you should really wish for. If you're trying to cause a fight to occur you could strife out the Doji courtier that you're opposing, causing him to break out into an impromptu poetry jam session and profess his undying love for his wife (because that's what he does when he breaks composure.), or you could simply incite that same doji courtier to take a swing at His usual nemesis, Ikoma Killsyuo, which is far more likely to get the result you wanted.
As for strife being "more important than fatigue", I would have to say that strife is more annoying than fatigue, but certainly not more important. When you max out on strife, you either unmask or you can no longer keep dice with strife symbols on them. When you go over your max endurance you literally can't take actions which require dice rolls. If you max out on strife and people keep hitting you with abilities to increase your strife, literally nothing happens. If you max out on fatigue and people keep hitting you with fatigue causing abilities, you start taking criticals. I can still see saying that strife is more annoying than fatigue, because with fatigue you're either still in the fight, or you can go play on your smart-phone while the rest of the group keeps playing L5R, but with strife your forced to keep paying attention and trying, even though your chances of succeeding at an action are greatly reduced.
Along with water and void stances being used to combat strife, I believe it was earth stance that allows players to remove strife from other targets, due to the stabilizing demeanor of the character using earth stance.
I completely agree that strife should be a bigger part of dueling. In the lore I remember that doing some pre-duel show-off stuff is supposed to be a common part of dueling and I believe this would be a great time to allow a duelist to use a strife inducing action of some sort. Keep in mind also, that all of the various conflict types allow for "special actions" which could easily be used to take some sort of strife management action. Furthermore, I personally would encourage other players to attempt to influence a duel by using subtle strife management actions. (This is largely because I feel like other players should have something they can do during a duel.) I picture the story with Akodo Toturi and Bayushi Aramoro dueling for the emerald championship. Before the duel starts, Aramoro and Toturi both show-off.. Toturi's pre-duel action is specifically noted to be a subtle insult to Aramoro, which probably would increase Aramoro's strife for the duel. Meanwhile, Aramoro has a helper in the crowd who is shining a light in Toturi's eyes, showing how people who aren't actually dueling might still have some influence over the duel.