Fate is a vague definition. Remilia's ability itself is vague too, what can she do to fate exactly? Can she just change fate at an exact point, like, "She will get hit after 3,2366845 seconds." or she has to make up everything to go to that fate? Like "in the next 0,812387 she will step right for 82 degrees, 5.01829 centimetres,..."
Because fate is not an exact thing, people can change their fate, or so they think. Knowing the fate will surely change it, though. Therefore, Remilia cannot change the fate as the latter example to get a perfect result, one fate involves other's fate, a slightest change in your fate may greatly affect another fate that collided with it.
Ex: I randomly discovered Touhou and got too obsessed. My otaku level went straight up which wouldn't happen if I never cared about what one of my friends said about Touhou. My friends wouldn't have to hear my rant about Touhou everyday. My HDD wouldn't have 2TB of Touhou so I may can store something else more useful. My parents wouldn't love The Gensokyo the Gods Loved and have History of The Moon, Broken Moon as their ringtones. MY BROTHER WOULDN'T BREAK UP WITH HIS GIRLFRIEND BECAUSE OF SAIGYOUJI YUYUKO.
Remilia may have absolute control over fate, but that consumes too much effort, time and her own magic power, so she wouldn't do that randomly just for victory of some stupid danmaku game.