1998 Honda Accord Cleaning oil from the coolant system

A while back the mechanic noticed an oil in the coolant. He said that I need to replace the head gasket. I also did some search to see the cause of that and people also
pointed out that it is very likely it is the head gasket; oil level was going down. I replaced the head gasket, oil level is good all the time.
I also did complete system flush that cost me $95. I asked the mechanic, is this going to completely clean the system from the oil? He said no. Then I asked,
will that affect the engine in any way? He said no. I was not convinced. Until now I still have oil in the coolant, I flushed few times and cleaned the overflow
reservoir where the oil usually accumulates and I still do that hoping that one day all the oil can come out. I found forums that say oil in the coolant can
cause damage to the hoses.

The car has no fluid or oil leaks, what do you think about this situation and I am wondering if you have a suggestion on how to get the cooling system cleaned up?

Thank you for your help