Not sure why you need distilled water. The point of the pumps is to oxygenate the water, which the roots of your plants like, you don't need both those heavy duty pumps to aerate some water though. I'm sure bubbling the water helps the chlorine get out of it too, which is the main goal because chlorine is gonna wipe out vital bacteria in your soil.
Doing this (bubbling) to your tap water in conjunction with leaving it in an open container, like a 5 gallon bucket, will allow the chlorine to evaporate out of the water and make it safe to use on your plants. I think that's what your getting at at least. I'm not sure if the water will be ph balanced, soil rich in humus will buffer any ph imbalances though. Bubbling tap water in an open container for 24hrs is how you get the chlorine out of it and get it nice and aerated before you water or mix nutrients in it.
Additionally, to distill water, one doesn't simply "boil water". You actually need to boil water and collect it as it turns to steam.