I do hunters/jumpers and eventing. Here is my take. In the hunters a PROPERLY adjust standing would not affect performance. (IE the head set) hunter should go around soft and flowy with their nose poked out just in front of the vertical. I have seen plenty of horses learn to lean on a standing and even more horses go around with too tight standings. The only time a standing should come into play, is when the horse gets it’s head dangerously high . There are also many horses that learn that if they suck back, they can lift their heads higher. Since hunters max at 4’ (for the most part, only one division has the option of going to 4’6), there usually isn’t as much stretch and reach as you would see over the wider jumps in the jumpers/eventing. In the jumpers after 4’ or prize money of 15k (I believe don’t quote me on it) standings become illegal and runnings become the norm. There has been a discussion about changing the rules to allow runnings in the hunters. The hunters are ridden on a much softer and lighter contact. When I ride my hunter, I found that the running barely did anything because I spent so much time on a loose gappy rein.
For the eventing/jumpers, where you have to take make more precise movements at (often times) a lot quicker pace with a much more keen horse, I prefer the running. Again, it needs to be properly fitted. Would I feel comfortable schooling lower levels of XC in a properly fitted standing, yes. Would I do it? Probably not. A too tight running breaks the line from bit to elbow down, thereby ruining the communication and causing constant pressure on the mouth.
Basically, it all boils down to what your horse does better in and whether or not you have learned how to properly adjust the equipment. WHY you are using the equipment is also a very important piece and these pieces DO have a specific function. I have seen plenty of too loose standings and runnings used just because people thing it completes the look.