Yes, the water, etc. most likely was from the old radiator, not the new one. If the transmission specialist said the transmission was fine, talk to him about doing one more drain and fill. Two drain and fills would have gotten out about 2/3 of the fluid.

 

I'd just drain & fill a couple more times until it stays clean. Normally we say 3-times, but that's for old dirty fluid. If it was my car, I'd want to make sure ALL the water was gone.

Drain & fill is pretty easy to do yourself. Wanna try that?