I consider myself to be a (fairly) serious griller/smoker, but I've never used chestnut wood as a smoking medium. It's a hardwood/nut-producing tree, so it should qualify as a viable option for smoking, but I couldn't tell ya how the food would turn out. I certainly agree that if you burn a small amount if it, and it smells acceptable to you, then it ought to produce an acceptable "end product" if you smoke with it.
Since your chestnut tree is freshly cut, you might want to immediately sample the smoke properties (rather than waiting anywhere from 6 months to 2 years for the wood to fully cure). If you have an electric bandsaw, or a circular saw, just cut up a couple of larger pieces into little chunks....and save the sawdust. You can dry out the sawdust in just a day or two; then smoke a small amount of food with it. That'll be an accurate "preview" of how the wood will behave.
Here's a helpful site (usually) that I've relied on for ages. They don't have that much to say about chestnut, however!