OK, if you apply a small slow-drying paint daub on the center of the tread on a rear tire and roll the car forward, the paint daub will make two marks on the ground, measure between them and this is the actual rollout on the tire.

If the tire is 85 inches in circumferance (the rollout) the working diameter will be 27.056". If you use your "lay a stick on top of the tire and measure to the ground" you will likely get a smaller measurement of the diameter than the actual 27" due to that flat spot on the bottom of the tire but the rollout will still be 85" and that's what the odometer needs to see.

If the flat spot is 1/2" your measured diameter will be 26 1/2" which will make the rollout 1 3/4" too short which will result in a 2% error. In the real world you'd be lucky to come within 2% accuracy from one pinion to the next larger/smaller size but the point is the "stick on top" measurement is not the correct diameter for computations especially considering that many tires will have more than 1/2" squat.