A “Cinchy” horse is fairly common for many reasons from having skin pinched at some point making them sour to sore backs/withers and everything in between.
One way I have found that works well for figuring out if the horse has an issue with the girth area is; take a soft lead rope and put it around the girth area of an unsaddle horse and pull tight slowly with your hands (DO NOT TIE). If the horse has no signs of being ” Cinchy” then there is most likely an issue with the saddle pinching or the horse is sore somewhere other then the girth area.
Another common mistake people do is cinches a horse up quickly as tight as they can all at once, this can pinch skin or just cause discomfort in general. Take your time start with slightly snug and cinch up a hole or two several times before mounting.
You can also gently stretch front legs forward to help prevent pinched skin. When stretching legs forward keep low to the ground and do not pull just support let the horse stretch against you.
Hope this helps!