is it actually necessary to remove the engine belt?
can the process be done (all the steps mentioned) without removing it?
Not necessary unless you find it hard to undo the bottom radiator hose at the engine.
*My 2c Worth...
You should bleed the system from the bleed screw on the thermostat housing while the thermostat opens with the radiator hoses all connected...
You should also let the system reach full temperature and pressurize...checking for leaks in all cooling system connectors.
Check that the over flow bottle is at least half full...Then once the cooling system has reached full temperature and pressurized, you should place a rag over the radiator cap and twist it open until it hits the first safety stop in the half locked position (if you don't know how to open a hot radiator cap safely be warned you may burn yourself at this step with scolding hot coolant - so practice opening and closing the cap to the first stop until you know by feel how to do it). This step will force coolant out the radiator up the the overflow pipe and into the reserve bottle blowing out all air in the overflow tubes in the process. This ensures that once the engine cools down and thermal contraction occurs, any needed coolant from the reserve bottle is sucked in by the cooling system instead of sucking in trapped air in the overflow tube. :thumbsup:
Ensure the reserve bottle is filled to max after this step and check it again when the engine is completely cold to see how much coolant was sucked in...If the bottle goes under minimum you didn't do such a good job bleeding and will need to repeat the process starting with bleeding at the bleed screw.