Or does Congress only have that power when it's something you agree with?
I'm not sure which part of this you think I agree with. I certainly don't agree with what the FCC is doing here.
It sounds as if you're challenging whether Congress even has the authority to create independent agencies in the first place, which is an entirely unrelated discussion, and one for which I don't have a particularly strong opinion (I've just taken it as fact that it's something that they do, without ever really questioning it, honestly). That said, I will point out that if you take issue with Congress delegating its authority to independent agencies, then you'd need to account for the existence of NASA, the Smithsonian, the National Science Foundation, the Federal Trade Commission, the Securities and Exchange Commission, the National Transportation Safety Board, the Consumer Product Safety Commission, and even the United States Postal Service, all of which are also independent agencies. Again, I don't have a strong opinion, but there's clearly a necessity that those services be provided in some form or function, and the 538 members of Congress are clearly not up to the task of managing all of that on their own, especially once you consider that most of those agencies are far larger than Congress itself.