The reversing switch needs the ball bearing to function. The switch plunger pin won't make contact with the selector rod. If you think the radiator is still clogged with crap you can try an old school trick. Take the radiator out, completely drain it and lie it down with the outlets facing up and the rad cap on. Mix up a strong solution of citric acid powder and hot water and fill the radiator to it's capacity and let it sit for a few hours. Drain the radiator solution into a bucket so you can use it again if need be and if you have a pressure cleaner, blast it into the radiator bottom outlet to reverse flush all the dissolved debris through the top outlet. This won't harm the radiator but should be strong enough to break down sediment in the radiator to the point where it can dislodge from the cores.

*make sure all the solution is drained from the radiator and flushed out as, although this won't harm copper or ferrous metals, it will attack alloys and cast alu parts.