At some point there's just so much noise you'll probably find carcinogens everywhere.
If a passing electromagnetic wave were to cause current in the DNA right when the damage detection and repair machinery is checking things out that might interfere making it miss some damage, which could result in cancer.
In that case the electromagnetic radiation would not cause the cancer, but it would make some other thing that can cause cancer more effective.
You'd probably never be able to see this if you studied an average population looking for signs of increased cancer in cell phone users. Where you need to be looking is populations that have a high cancer rate, where their bodies are already near the limit stopping DNA damage from turning into cancer, and see if adding cell phones increases cancer rates.
Can a passing electromagnetic wave cause current in DNA? It should be possible. The real question is what wavelengths can do it. DNA is pretty small, so you'd expect that you would need a very short wavelength--several orders of magnitude shorter than 5G.
But DNA is actually a very long molecule that is twisted and curved to pack it into a tiny space. If it could be laid out in a line it would be around 2 m.
Weird things happen when you take an antenna and fold it into a smaller area, especially when the shape it undertakes has a lot of self similarity, including handling longer wavelengths than you might otherwise expect. Would that be enough in the case of DNA to increase the wavelength that can cause currents into cellphone range?
If I had to bet, I'd bet "no", but I wouldn't place a large bet.
The local heating one always seemed much simpler, particularly as you didn't even need to have the heating come from the antenna, as I remember old cell phones--which you usually kept next to your head while talking (how quaint ;P--also routinely getting hot to the touch just from heat dissipation from the battery. There were tons of other explanations for the effects people found, though (which were stuff like "while we didn't find more brain tumors, if you got one it tended to be on the side you held your phone").
That is to say - just because visible light is more energetic does not mean it is guaranteed to cause more damage.
(I briefly worked in Tor's lab - a truly incredible chemist and teacher)