I've taught engineering thermo and heat transfer in college and also worked on everything from watches to building fiberglass
highway bridges. The people are right that are telling you the heat transfer coefficient of the tube material normally has a very small effect in the overall temperature difference between the fluids on either side of the heat exchanger
walls. The transfer of heat from the liquid to the tube and tube to the air tend to be the controlling factors in this problem. The different coefficient will mean there will be a slightly larger difference in temperature from the inside wall to the outside, but not enough to make a big deal. For example there might be a 10 degree difference between average liquid temp and inside wall, a 1/8 degree difference between inside and outside wall and a 20 degree difference between outside wall and average air temp. Changing from copper to stainless, might make the wall to wall temperature difference be 3 degrees, changing the total from 30 1/8 degrees from water
to air to 33 degrees. Nothing to loose much sleep over.