Assuming as Tooljunkie pointed out there is a real need to clean the rad due to debris in your system my last recollection of properly cleaning a rad is to have the top and bottom tanks removed and then having the core cleaned or replaced as needed.
I also agree that a working temp gauge stock or cheap aftermarket would be very useful, just keep in mind if there is no flow the temp will not be reading correctly.
My gut is telling me that if the brass thermostat is that dissolved that the steel water pump impeller is gone as well and your freeze plugs and rad are suspect.
My assumption is that your engine is now working like a percolator coffee maker, boiling water and belching it out the overflow till there's nothing left to boil.
