Oil in Coolant Bottle

Hi

My 2010 Aveo is leaking oil into the coolant bottle. There is a brownish paste of the cap of the coolant bottle. In the coolant also there is a thick layer of this brownish oil floating. I did gasket change in the last few days, but the oil comes back after driving for a few kilometers.

It appears that the transmission oil is leaking into the coolant because I have noticed that my transmission oil gets finished quickly. Then I have to top it up every second day.

My mechanic suggested that the problem is in the radiator(since my engine doesn't have an oil cooler). Transmission cooler located inside the radiator is leaking mixing coolant and tranny fluid. I have taken the radiator to a specialist who did not find any issues with the radiator. Now I am planning to take the radiator to another specialist who has been in the business for decades. I don't mind buying a new radiator, but it would be cheaper to fix this one than buy a new one.