Why does AgCl exist as a precipitate in water?

So you're thinking of all salts as being completely soluble and just to get it out there, that's not quite right. This can be seen in the solubility of AgCl with, like you said above, a Ksp=1.7*10^-10. A better way to think of it is that there are a great range of solubilities for different salts and for any compound for that matter. NaCl would have a very high solubility because you can add over 30g into a 100mLs of solution. Another solubility that's pretty high is sucrose, you can add a whole load of that to water and it will dissolve, even if it takes a while, and that's not even a salt. Now there are species on the other end of the spectrum, like the silver chloride, or even organic compounds (which are non-polar). Even something like hexane will have some solubility in water. If you take hexane and put it into water you will have two phases because the two don't mix well, but if you remove the hexane from the water and then smell the water it will smell like hexane and that's because there is a little 'left over" in the water. (I would caution you with the hexane though, it's so low that most people consider it to be immiscible, just like most people consider silver chloride insoluble in water.) You can also find substances that are in the middle with lower Ksp's, that many might call marginally soluble. Check out the Ksp's in the source, things like lead (II) chloride, cadmium fluoride, or barium bromate. I hoped this helped. If anything just try not to think of things as completely soluble or completely insoluble, but more as a spectrum. (Frustratingly things are usually more gray than black and white...)