Hardly. You're making the argument that our rights should be infringed for the public good. If we use your reasoning, then every right granted to us by our Creator is at risk for the same reasons, including our right to keep and bear firearms.
The 1st Amendment does not give people the right to be profane.
Nothing in the Constitution grants any rights to anyone. The first amendment says that Congress may not abridge our freedom of speech, among other things. How is the FCC's disallowing of certain content fitting in with that? It doesn't.
The original intent of the FCC was to regulate airwaves so that radio and later TV stations would not interfere with one another. As these airwaves span state lines, this is a perfectly reasonable and legal use of the interstate commerce clause of the Constitution to regulate who uses which frequency. This cannot, and must not, twist into the ability for the FCC to limit free speech.
You're on TV or radio? Tell me, what's the point of licensing broadcasters if anything goes? If anything is permissible? When their licenses are up, they have announcements that tell viewers they can make comments about renewing their license as to whether they serve "the public good". Why?