Common mode error formula from TI seems counter intuitive
The common mode error appears as a delta in the offset voltage of the amplifier so even when you eliminate the common mode component of the error some offset will remain. 

In short if you avoid any changes in the common mode voltage and match the design CM voltage, there will be no error contributed by the changes in the common mode voltage. There will still be other error sources. 

In general, in some, not all, cases, the common mode voltage of the amplifier does change significantly so we need to consider the CMRR. 

For example, if you have a high-side shunt and you are measuring the voltage across the shunt, when the output voltage changes you will have a change in common mode voltage.