You'd be surprised by how much water has been saved in droughts by encouraging this kind of small scale efforts. E.g. the Cape Town water crisis
Like the OP, I think one should analyze the problem before opting for the "obvious" solution. I did so with water a few years ago. IIRC, for individuals the biggest share is for showers, then toilet. These two account for two thirds of standard use.
If you encourage people to turn off the sink more often, you may gain 20%... of 5% of their average water needs, so 1% overall. If they take fewer showers, each with less water, they easily can halve their consumption, so a gain of more than 20% overall.