I put mine through the rivet. That way I can close the vent without removing the thermometer. As far as taking temp at the grate versus the lid, I use a pretty long stem on my thermometer. That way I'm reading pretty close to where the food is. If you mounted it anywhere on the lid, you can get close to the food by using a long stemmed thermometer.
One other thing I like about the thermometer through the rivet is that I can easily adjust the thermometer if the meat is in the way (like with a full slab of spares would be).