You do not need to remove it. Removing the relay just makes it stop working. We have procedures from the factory we follow regarding the AIR pump, we connect to the car with our computer,use our GM software to update the car computer and eliminate the program from the computer then attach a required decal stating we did a federal emissions update to delete the AIR pump device.
Removing the air pump,bracket,tubing,check valves,hoses and capping off the manifolds are all things done for cosmetics and slight weight reduction of unused and unneeded parts.
It is legal to DISABLE the air pump per the TSB. It is technically not legal to remove all the disabled hardware. It is a fine point but you may fail an emissions visual check in some areas if the hardware is removed. (This is not a problem in Ohio.)
You can get an official looking sticker from an online vendor that says all the hardware removed was disabled per the TSB. That has allowed some owners to slip through those checks. (Legally it means nothing, but some inspectors have accepted it in the past.) Otherwise, you will have to reinstall all the hardware.