Unless the engine seized up and doesnt even turn over, or unless the engine has a loud, deep knocking noise, or there is little to no compression in the engine, a mechanic wont know if the engine is good or not until it is torn down and the internal parts inspected.
With the engine overheating to the point that the cylinder head is warped, it is possible for internal engine damage such as a cracked block, spun main or rod bearings, thrown rod, or broken crankshaft.
They are correct that you cannot tell about other damage until the radiator is replaced as there is no way to know if the engine is still going to overheat from a blown headgasket or warped head unless the cooling system holds coolant
Because you dont mention any knocking noise that I asked about, I assume no engine knock when it was running between the repairs. That is a good sign. It is also a good sign that the engine was running.
I suspect that there is another issue with the engine, possibly a crank or cam sensor, overheating fuel pump or other issue which is causing it to run for awhile and then stop running. at this point, you need to have another shop inspect and test it as I do not believe that the engine is bad