We tend to carry all of our water now plus extra (actually, read that as I tend to carry the water, my girlfriend carries the food - not sure about the arrangement there…).
We tried using the chlorine type tablets in Morocco 2014 when climbing Toubkal and we had no choice but to use stream/lake water. I ended up being a bit ill, which my GP thought may have been Giardiasis:
Having recently spent a couple of weeks climbing some >2500m mountains in the Swiss Alps, I know what you mean about using up the water quite quickly. Luckily a lot of huts do sell water, which you can plan for using hiking maps and a bit of Googling to check they are open (you may also need a lot of cash - my worse bill was when I spent almost £50 on 5x bottles of 1l water on a long distance overnight trip)!
We also take a small gas stove and 100g gas canister, just in case we need to boil some locally sourced water. You can’t beat a hot cup of tea on a summit as well!