Your block may even be plugged for all I know.
This IS a very good point! IF the radiator needed to be cleaned out, internally, then the internal cooling passages of the motor might not be far behind. Remove ALL of the block "core plugs", rinse it all out, especially the rear passages (the lowest part of the motor, as it sits in the car), which will be messy to do, put some new brass plugs in, and things should be better.
A "short-cut" way might be to put some flush solution into the cooling system, run the engine for a while, then drain it out. Take the thermostat out, plus remove the lower radiator hose, and rinse things through the thermostat passage. That might get some of the residue out, but it might also remove any "scale" that's been covering any pending perforations in the heater core or existing core plugs in the block. Might end up taking some of them out anyway! Of course, rinse the radiator, too!
After doing this, then put "water" back into the system and run it a while, to check for any leaks and such. Drain and flush, then add 1 gallon of anti-freeze into the next refill, as anti-freeze will leak from places water usually won't. Think "pin holes". Then, before winter comes, drain 1 gallon out and replace that with one gallon of full-strength anti-freeze.
This will not be a "quick or reasonably easy" fix, but it might be the best long-term alternative.
Of course, dispose of such hazardous waste in a responsible manner.
Just some thoughts,
CBODY67