I use a garden hose spraying from the inside next to the fan forward. Lots of dish soap and a soft brush. It's easy to get the outside of the fins looking like new. A pressure washer could bend those fins & cause all kinds of problems.
If it's an old JD radiator it's worth saving and rebuilding. See if there is a radiator shop locally who does that kind of work. It takes a real craftsman to take one apart, clean it, and put it back together but it will be better than anything you can buy.
If that doesn't sound good to you, I've used Prestone Radiator Flush. It's an acid in crystalline form that you put in the radiator for 15 minutes and then flush again and again with a garden hose for an hour.... If done correctly it is as good as a rebuild. Incorrectly means you just bought a radiator.
To check if the OEM gauge and thermostat is working correctly you can simply run it without the radiator cap. Check the water temperature with a candy thermometer. When idleing & not working hard, the engine thermostat should keep it between 190 and 200F so it is well below boiling and the cap is not necessary. If it climbs past 205 shut it down and replace the thermostat and do the radiator cleaning inside and out as discussed. I've seen leaky water pumps, but never seen a water pump fail to move sufficient water. It was always something else. If the thermostat and cap are working right, most engines will do a density thermocycle circulation at idle enough to keep below boiling even without a water pump rotating at all.
To check a thermostat take it out, read the degrees printed on the top, and put it in a full pan of hot water along with a candy thermometer. It should open up at about 190 to 200 degrees. If it doesn't open until 220 there is your problem.
All the pressure cap really does is enable the engine when working hard to exceed 212F for short periods without boiling over. It sounds like yours is working.
luck,
rScotty