Toyota Repair: Overheating, radiator hoses, radiator cap

Question
The radiator cap bottom parts broke off. They are a 1/2"diam washer, a spring and another 1/2" diam plastic parts. Does the engine water jacket have small orifices that could be blocked by this pieces of debris and make the engine overheat? I found the spring by removing and flushing the radiator but couldn't find any other pieces. I hoped to find pieces that could be trapped by the thermostat but there was nothing there. I am about to replace the head gaskets and resurface the head, should I do that?
4runner 2000 4 cylinder. radiator cap interior components came apart while driving the car, antifreeze leaked out through the plastic reservoir, the engine overheated until check engine light came on and I stopped the car. I refilled with water after cooling down the engine and went home about 4 miles away, now the engine is overheating, I replaced the radiator cap, replaced the thermostat, flushed the radiator, removed radiator hoses and tried to flush the engine water jacket with garden hose trying to remove any loosen radiator cap components, put everything back together and still is overheating. No leaks detected. Is it possible that there are radiator cap components and there is a blockage somewhere in the engine.
I filled up radiator with water all the way to the top trying to see bubbles or air coming out of the radiator while warming up the engine and nothing. "I noticed that the car overheats while driving but cools down slowly when idle" does the same with or with out thermostat.
Water pump seems to be working OK, belts are tight, fan clutch seems to be OK.
What's the problem, how to fix it?
THANK YOU.

Answer
This is typically a headgasket problem, I don't think those missing washers can cause a blockage big enough to cause overheating especially since it seems to overheat mostly while driving and not idle. Those washers are too big to go through the radiator core which has coolant flow from top to the bottom, they may have traveled through the top radiator hose when the engine was not running they should not be able to go through the thermostat opening and even if they did they're probably in the intake manifold water housing.