The tiny amount of copper ions that would be in the water would take years and years for a bird to drink enough to be poisoned. Also if you change or refill the water often, there is even less in there. (stagnant water that sat for weeks might accumulate a measurable amount of copper but in a small bird bath it would evaporate)
A lot of people put pennies in their bird baths to keep the algae from growing. I also know a lady who has an antique copper bird bath that she's used for at least 25 years, and she also feeds the birds. She reports no problems that she has seen. Not that you'd necessarily know if a bird got sick from using your bird bath, I guess. There's a thread on this discussion about pennies in bird baths. The two articles linked have no reference to copper being toxic to birds.
Copper used to be (and still is) used for water supply pipes in your home. It's also used as a spray fungicide on fruit crops and other trees, so you'd think that would be a much more likely source for copper poisoning of birds than bird baths.
I wonder how researchers could come to the conclusion that there would be a problem? How can you do a controlled study on wild birds? Lots of birds only live a few years anyway, so . . did this bird or that bird die of copper poisoning? You'd have to find the dead bodies around your property and have them tested for toxic levels of copper to come to that conclusion. There are so many other possible contaminants, even in the rain water, how could you know?