I used hydrocortisone as part of the stuff I used to control it for years and I was fine. That is one of the least potent ones. I had my eczema sussed by the time I was about 11 and just managed it pretty easily from there. I always used Oilatum bath emollient, every single night, along with standard creams like Diprobase, and then used hydrocortisone when a had any kind of flare up in an area. Most important thing with eczema is to keep it under control and that means using more powerful stuff if your regular routine can't keep on top of a flare up, and then ceasing use when you have it back under control.
imo, that website is just classic stay at home mom psudo science, Fox news is one of their resources, most of their articles are over 10-20 years old, there are like 5 of them. Its founded by Dr. Marvin Rapaport and is based on his own research of which the most recent article listed as published is 2003. Its not something which has gained much traction at all, and their excuses for why are valid for pretty much any medical study.