FCC's job is to make and enforce policy that conforms to the law set by Congress, not to hold and obey public votes.

Yet, they are not allowed to ignore the public's comments. They must take them into consideration, even if they end up deciding to act against public opinion.

Ignoring what Americans want is almost the entire point of regulatory agencies like the FCC. They're a bureaucratic/technocratic check on the democratic process. (Which is why, in my opinion they're unconstitutional, but that's an argument for another day.)

That doesn't really explain the fact that a lot of agencies were created because of public outcry, such as the FDA. I think agencies exist because lawmakers don't really have the ability to move decisively when it is needed.

isn't the constitution itself a check on the democratic process?

Many mechanisms in the constitution serve as checks on the democratic process. Adding checks in addition to those is unconstitutional: http://www.law.nyu.edu/sites/default/files/ECM_PRO_060974.pd....