Fought the same issue several years back I had always used air to blow it very clean then washed it out with lots of water. I think the bean fuzz gets wet and packed in the fins in the bottom and like you said gets packed in there. Replaced thermostats and water pump but was still running warmer than I wanted. Was going to remove and have it boiled out but with the amount of work to remove just replaced with a new one rather than risk having to do it again later. Seems to have cured the issue on mine.

Radiator Genie tools for air and water might help you direct the flow into the fins better but probably won't help the packed and hardened dirt. Also good to use distilled water when mixing coolant to avoid hard water deposits which can clog radiator passages over time with scale, rust, lime deposits.