There are several paint coatings that have been used on concrete pools and the two most common are Chlorinated rubber paints (horrible coating that has to be re applied every few years) and epoxy paint, there are good epoxy coating and bad. One Lady contacted me last year about one she had used and she was able to get a refund as it didn't do what it said on the tin but she had to agree to a gagging order so the Company doesn't appear to look bad when searching the net. I have the name somewhere but will have to look it up as it's on my old PC.
There is no calcaire in Chlorine unless you have been using Calcium Hypochlorite as the sanitiser. What hydrochloric acid does do is dissolve some of the calcium from the cement/concrete surface, it will soften the surface.
Depending on what the previous coating is depends on whether you should do the whole pool as trying to bond to old coatings can be an issue. It's generally better to sand or sand blast the old coating off.
"the paint isn't bubbling or cracking but some flaked off" So it's not worn off, that means a potential problem area which should be looked at properly first, Is the concrete porous, should a sealing coat be applied before any paint coating is applied?
Epoxy coatings are not normally thinned but as I said previously there are some companies products which may differ in application and result.