In my experience, a well maintained pool should not contribute to asthmatic reactions, regardless of if you're using bleach, tabs, granular, gas, or salt chlorination. it's the combined chlorine that mainly contributes to triggering the reactions. Many pool operators are not maintaining the combined chlorine levels to a low to zero ppm, and are especially worse at public pools, and during swim competitions that are not controlled properly.
One of the benefits of a salt chlorine generator system is that the combined chlorine is superchlorinated as it passes through the cell and the cell is energized. This helps to control the combined chlorine levels.
Tall - while the initial investment is usually higher, there is a significant cost savings between producing your own chlorine with a salt system, vs purchasing packaged chlorine, equivalent pound per pound. That is, if you pay $1.00 for a gallon of 6% bleach, I can make the equivalent amount of lbs of chlorine with the AutoPilot for $0.59 (assuming $0.10 per kilowatt to run the unit and a replacement cell cost of $900).

Remember, salt chlorine generators are viewed on residential pools as more of a convenience device. For me, I can't calculate the dollar value of this convenience factor, especially since I travel most of the time, and no one else helps to take care of my pool for me. I don't spend much time at all when I am home, to keep my pool clear, clean, and balanced. I don't have to worry about a family member, neighbor, or friend to add chlorine for me, when we are on family vacation. I don't have to worry about algae growth at all. Basically, I have Peace of Mind. For ME, that's worth the investment.