First, you do not have to remove the radiator. Go to the spray wash and blow out the debris that way.

"If" you have grease on the radiator core more so on the condenser (ac intercooler), good chances the AC condenser is leaking refrigerant oil. Condenser then will have to be replaced to keep the AC working properly.

You were already given the flush procedures.

 

Flushing the cooling system is elementary. First, you pour distilled water into the radiator. Then, start the car and let it idle for about 15-20 minutes. Next, turn off the engine and drain the cooling system. Repeat everything several times if the vehicle is relatively new. If there are no contaminants in the antifreeze, this method will work. You can use acidified water to remove scale and dust particles. But you can also use a special engine flushing fluid. I have seen this method at Commercial Pressure Washing - Professional Building Exterior Washing, which deals with high-pressure washers. But you can also try.

 

Use ZEP citrus based cleaner/degreaser. It's safe for metals and works great on the exterior of the radiator.

https://www.lowes.com/pd/Zep-24-fl-oz-Degreaser/1000207949?cm_mmc=shp-_-c-_-prd-_-lwn-_-ggl-_-LIA_LWN_127_Cleaning-_-1000207949-_-local-_-0-_-0&gclid=CjwKCAjwo7iiBhAEEiwAsIxQEdmuSIwt36b-oiiXCtz_ABK7LXqxpt2Ho_70F0XEsy9RVp-lh5GWgRoCvQEQAvD_BwE&gclsrc=aw.ds

Here are some pics of my radiator after cleaning it while I had the old AC condenser w/integral oil cooler out (for replacement)...https://www.gmt400.com/threads/condenser-oil-cooler.59140/post-1260762