IMO there's nothing wrong with an automatic water change system, but like DavidR at what point does new water become old water? From what I perceive is an effective automatic water change system is that it dilutes the nitrates almost immediately and then new water becomes old and what ever goes out is old water. I personally am looking into investing into an automatic water change system, because when I'm off to uni, I'm not going to be able to lug a 500L tank down to Dunedin and the parents are getting too old to carry buckets and buckets of water.
Those equations are quite simple
Day 1
100/100*86 because it was 100% old water and is now 86% old water due to a 14% water change. To work out new water simply go (86-100)*-1
Day 2
You start off with your 86% old water
86/100*86
Ect and you use the days before (% of old water divide by 100 and times by (100 minus how large your water change is))
Not saying that your method is ineffective but having a hardy turtle and 4 danios, I think they can cope with a degree or two in change of temperature, a small pH fluctuation (.5?) and I have heard that nitrates aren't highly toxic until 100ppm+ so they an cope from 5-20ppm a week. But I agree with you if you have sensitive or non hardy fish manual water changes could stress them out.