The reason that not many use a Sony (although it's probably a good camera), is that they do not have the huge system, and range of lenses that Canon & Nikon have accrued over the years, and there is only two choices for professional shooters.
....Use the exposure compensation and select a negative value (depending on the situation it can be -2 to -3) if the overall light is rather dark.
There may be some situations where this would give you passable results, but normally it would (as you say) under-expose the whole image. This might work if you have spotlights on the subject but if you have backlight it would be a complete disaster and probably pretty terrible with balanced front/back lighting. I would have thought that gymnastics or any other indoor sports would be lit fairly evenly and constantly lit (however bad the lighting) and that you would normally get the best results by shooting on manual with settings which suit the particular type of shot you are shooting and the limitations of your camera and technique. However good your focusing system it will be much easier if you can shoot fast moving subjects from side on so the subject to lens distance is constant.tobFrom my personal experience in shooting ballet on stage I can give you the following advice: - Use a camera that has a really good autofocus and is able to focus with low light, e.g. Canon 1Ds MKIII / Canon 1D MKIV / Canon 1Dx / Canon 5D MKIII or a Nikon equivalent. - Use lenses with an open aperture of 2.8 or better, 24-70 / 2.8 and 70-200 / 2.8 IS are really good for most situations - Depending on the available light you might need to choose ISO up to 6400 to get a decent image - Use shutter priority at e.g. 1/125s or less and disable safety features that prevent taking the shot if it seems to be underexposed - Use the exposure compensation and select a negative value (depending on the situation it can be -2 to -3) if the overall light is rather dark.