No, a slave flash would increase the available light. The ISO deals with sensitivity, a higher ISO with a big flash will increase the effective distance the flash can illuminate. These are complex concepts, in general terms higher ISO will increase your ability to use higher shutter speeds and capture action. Image stabilization is useful for eliminating the effects of camera shake for hand held photography but it will not help capture action, instead a higher ISO and a monopod (or tripod) for support (to eliminate the camera shake) will help stop action and reduce motion blur, a more important issue if you are shooting sports.
DSLRs are the ideal solution, but a DSLR with a 70-210mm f2.8 zoom will cost nearly four times your budget. Using a flash will not help unless it is a real big one (high guide number, there we go with more numbers again) since the effective range of most flash units is less than 20 feet, not much use for any sport beyond chess.
IMHO you should look at the cameras mentioned here, try them out in the store if possible, and I think the Fuji S5200 will be the closest to what you need. Fuji EVF cameras tend to have viewfinders that do not black out during burst mode as well, another plus for following action.