Not too worried about the chlorine. It does it's job, then dissipates, much like sulphite. A dog won't touch a bowl of water straight from the tap but leaves it till the chlorine smell has gone. The feed from the water main to my house is lead. When the goverment initially tried to sell off the nationalised water system they had no takers because it was uneconomic to guarantee water to the standards stipulated. The safety limits were reduced by a factor of 4 overnight in order to make the sale. My water has a pH of 7, ie alkaline, and when I got an analysis, (Thames Water's own, not independant) I bought a water filter. Due to the drought, they are now using a desalination plant and extracting brackish water from the the Thames estuary. Ok for flushing the toilet. Rainwater is not much better. Every year I have to clear the gutters and what comes out makes good compost (nitrates etc). Likewise the sediment in the water butts.
But even my local grocery store, not noted for being cheap, is now selling mineral water for £2 per 10 litres and it's better than my jug filtered tap water. Fermentation, fining, filtering and ageing, especially in oak or PET containers, probably does a lot to transform any water used in the process, but as all kits require the addition of water (unlike real wine and good cider) at these prices for mineral water, for the expensive kits which require less water, the choice is obvious. For the cheaper ones, using cheap bulk mineral water only adds £5 to the cost of 30 bottles. I'll try it next time.