I am a little confused.
I did not say two perpendicular wheels, I meant two parallel wheels like this (ignore the back wheel).
What I am understanding you saying is to just use the one horizontal wheel (labeled back wheel).
What I was saying is using just that one encoder to measure angle must be inaccurate. Depending how the robot turns, especially if it is not consistent (affected by dynamic speed/weight), that wheel will not always spin in proportion to the angle of the robot . If this is where I am wrong, feel free to correct me.
For example, if for some reason the robot was pushed so that the robot rotated directly around the horizontal wheel, causing there to be no encoder movement, would it not lose all accuracy? If there were two parallel wheels like in the diagram, it would not matter what the center of turning would be. And this will still be much more accurate than integrated sensors due to wheel slip and inexact point of turning (within the width of a large wheel).
Finally, in odometry the horizontal tracking wheel is not used for angle. It is impossible to differentiate the movement of the horizontal wheel between horizontal displacement or rotation of the robot. What tracking algorithms do is measure the orientation of the robot using the two vertical wheels with this formula:
dRadians = (dLeftInch - dRightInch) / chassisWidthInch;
Then, it can calculate how much it expects the horizontal wheel to move given that rotation.
It cancels out that movement from the horizontal wheel, and what remains is the horizontal displacement.
If all you had was the horizontal wheel, it is impossible to differentiate horizontal displacement or rotation. What if you had a tall stack at the front of your robot that made it so when the robot turns it does a slight arc around the stack? Then, the wheel would be spinning much more than usual for the same angular rotation of the robot.
If I am missing something important please correct me =)