You can definitely put a whole stick of cinnamon in with your coffee grounds or tea leaves. If your cinnamon stick is very tightly coiled, the hot water may not reach the inside of the coil. Split the stick lengthwise so the water can more easily reach the inside of the bark (that way you get flavor out of both sides of the cinnamon bark instead of just one side.
You're correct that you won't get all the cinnamon flavor out of a cinnamon stick in a single use. However, it doesn't have to be wasteful - you can reuse the cinnnamon stick multiple times. After your coffee is brewed, pull the cinnamon stick pieces out of the grounds, tap off any remaining coffee grounds, and lay them out to dry. It helps if you do this while the grounds are still hot, so the residual heat in the cinnamon stick helps it to dry fully. Make sure the cinnamon stick doesn't end up sitting in water. You can put it on a dry towel, or prop it up on the edge of a clean mug, or put it in a mesh strainer.
Re-use the cinnamon stick as often as you make coffee. When you notice it starting to lose flavor, break it into more pieces. Eventually, the parts will run out of cinnamon flavor and/or get too small to be worth the effort of saving, and you'll start a new stick.
Note: this works because the cinnamon stick has only been exposed to coffee grounds (or tea leaves) and hot water. Don't save a cinnamon stick that you've used to stir a cup of coffee with milk or cream. You won't be able to get all the milk or cream off of the cinnamon stick, and it won't be food safe for your next use, and it might taste bad. (Of course you can reuse it if you make your next cuppa immediately; just don't save it for tomorrow.)