Let the car just idle til comes up to temp. Make sure the fans are coming on at the proper temp settings, so when your in traffic, you know there actually coming on. With NO AC running the fans come on at low speed when the coolant reaches 226 degrees. Then both go to high speed at 235 degrees.
Your local auto parts store sells a product you spray into your radiator and a/c condenser fins. When this product foams up, it flushes all the packed dirt a bugs from the cooling fins, then flush with low spray water, can actually see thru the radiator and condenser after cleaning. Im sure you already did this, but the coolant level is normal? Yoy dont have the common leaking hose to the expansion tank, which could also introduce air to the system.
I thought my radiator and condenser were pretty clean, after spraying the cleaning product into the fins, when it foamed up, tons of crap poured onto the ground. I was like, " damn". Hoses it all down, cherry!
Industrial air conditioning techs, like Trane, Fujitsu, etc also use these products on large air conditioning units. Should only take a few minutes to disassemble the fan shroud.
I you have an air compressor in your garage, you can make a fitting to fit the hose, copper tubing, like Jb is refering too, then bend a 90* tip on it, but you should still spray the fin cleaning stuff in there, the air alone wont clean it as well as the cleaning product used for this purpose.