Water quality and diet are the most important issues, but carbon can have the negative effects described. Many people who have plants who stop using carbon notice an improvement. Carbon won't kill your fish or plants, but without it they can do noticeably better. Water clarity should not be an issue with adequate filtration (not including carbon) and water changes, the same goes for any smell.
And yes, there have been cases where carbon has been shown to be associated with HLLE. When the carbon was in, the HLLE developed. When carbon was removed, it went away. When it was returned, so did the HLLE. Every case is different. Most are not associated with carbon, at least not as a significant cause. But sometimes it does make a difference.
Carbon does not just remove smell, discoloration, and medications. It removes all sorts of chemicals in the water column. This does include things like trace elements and micronutrients.
I don't use it and have seen an improvement. I have not had any problems and have had less issues with illness. Maybe others have a lot of reason to keep using carbon, but I do not see any. In my opinion and experience the same benefits you can get from carbon can be achieved with cheaper or free water water changes without running the risks of the consequences of carbon. Water changes are needed and good no matter what, why waste money on carbon?
Diet is not just what they eat, the major nutrients such as protein, carbohydrates, and fats. It includes many other things like the vitamins mentioned as well as micronutrients and trace elements. Some of these are provided in the water, from which carbon will remove them.
Please note that I am making a distinction between HITH and HLLE as in HLLE does not include Hexamita spp. or Spironucleus vortens infestations as the only or main cause as opposed to HITH where these parasites are present (and the only or main cause of the ailment). However carbon can contribute to both.