Silicon Dioxide (sand) is the mot common compound in the earth's crust, and can be found in virtually all soil types and exposed rock to some extent. There is no reason at all that it could not be extracted from a given soil layer.
The process is pretty much like the OP describes.
First you wash it. The finer particles are much easier to keep in suspension via agitation, and most of the organic material is water soluble. Rinse, Repeat, Rinse Repeat as needed. Then, as the OP said, you end up with a sludge. You could probably use that to make the glass (as the furnace is hot enough to burn off most impurities) but an intermediate heating, followed by a washing to get rid of the ash, would result in a more refined product.
As for my suggestion:
- Any river,lake, brook, cave river tile is a valid sand source
- Any soil type with "sand" or "sandy" in its name counts as sand.
- All other soil types can be made into a bag of sand at the glass furnace with fuel, and a bucket of water.
- For the complexophiles the duration/iterations of the task could vary by soil type.
- Quartz is mostly composed of larger crystals of SiO2, so you can process any quartz gem into sand at a millstone.
The idea of trading for sand is at best a stop-gap measure, but one that I'd welcome. Normally I'd oppose modern knowledge of chemistry dictating in-game logic, but glass is in no way a balance issue. Almost everything you can do with glass, can be done with stone.