You gotta find that part. Sounds like it's the smallest rubber seal in the middle of the radiator cap? It WILL obstruct flow when it gets to the thermostat.

But... I don't think it can get down through the small tubes of the radiator. First thing I'd do is remove lower & upper radiator hoses. Use a garden hose to flush TONS of water backwards (upwards) through the radiator. Trap the water that comes out, so you can find that little thing when it squirts out from the upper radiator hose (or from the radiator neck where the cap goes). Don't leave the upper hose connected to the engine - don't let it flush that part back into the engine.