Okay, how about a realism check here.
Here's a saddle attached to a horse. It's an English saddle because it's easier to see. The black strap at the bottom, not many shades darker than the horse, is the girth. As you can see, the girth fits snug to a horse. When riding, an English girth should be snug enough that you can insert your fingers between girth and horse but not your hand. It's literally right against the animal's body. Now, try to imagine cutting that with a knife or, worse, a sword, without injuring the horse. While the horse is moving.
Now, add this. Here the horse has a rider. You can just see the girth here...it's that dark stripe just behind the horse's shoulder. What is between any assailant and the girth? Yup. The rider's leg. Which in a battle situation would probably be armored. In order to get to the girth, the assailant would have to go under the horse - and a trained war horse would promptly kick them in the head.
Cutting the saddle girth in the heat of combat is simply not feasible unless your assailant is practically superpowered or just that insanely lucky.
Reins are more vulnerable, but losing a rein on a very well trained warhorse would not be an issue - they would almost certainly be trained to ride mostly off of the legs and seat to free up the rider's hands to fight.