That argument is actually the same error that GP criticizes: Wavelength (i.e. quantized energy) matters a lot.
Being irradiated by hundreds of watts of direct sunlight is fine for a short while (and causes skin cancer in the long term!), hundreds of watts of microwave radiation for the same duration is already pretty bad, and the only good thing about hundreds of watts of hard gamma radiation is that you don't have to worry about the long-term effects too much.
With mobile devices, we're talking about dozens or at most hundreds of milliwatts of microwave radiation though; there might be some effect, but I personally doubt it would be significant enough to worry about – earlier cell phones used way higher power levels, and I'm not aware of any link to cancer there. And they've been around for a while!
Last time I read up on it, the heating effect of holding any metal or plastic slab to the side of your head dominates any additional heat transfer from the EM radiation, even for fairly deep tissues.
I don't believe that low frequency would have adverse affects but I would like them to _really_ look into higher frequencies (as we approach microwave) just so, you know, we actually know if it's bad or not.
None of this "oops arsenic wallpaper is actually bad" "oops asbestos is actually bad" "oops lead in fuel is actually bad" "oops microplastics could actually be bad" stuff.
And we do care about walking in sunlight, it's called sunblock/suncream. People that spend their lives in a lot of sunlight are at higher risk for skin cancer, have you never seen what someone's skin looks like after a career of working outdoors?
As a Kiwi I guess I'm just more aware of the Sun; thanks to the rest of the world we have a huge ozone hole above us so our UV levels can get pretty mad around 11:00-13:00 in Summer.
..oops sun screen is actually bad too (Octocrylene is the latest problematic newcomer), and would need a very thick application to get the claimed effectiveness.