> somehow regulates cable TV providers, despite the use of wired connections

They regulate broadcast TV. Those rules leak into cable TV because the originators generally want content that can be sold for broadcast in the future and is advertiser friendly. Cable operators are also often beholden to community standards imposed by municipalities they serve. The FCC isn't responsible for content restrictions on cable.

> despite the use of wired connections to customers, instead of radio transmissions.

The FCC also regulates interstate wire transmissions.

But ultimately, you're not quite getting it right. It's all RF, it's just that we sometimes choose a really shitty wire called "the atmosphere".

Where in the Federal Communications Commission's governing legislation does it say that they're only allowed to regulate things sent through the airwaves?

It applies to communications over radio or wire: “The provisions of this act shall apply to all interstate and foreign communication by wire or radio and all interstate and foreign transmission of energy by radio, which originates and/or is received within the United States, and to all persons engaged within the United States in such communication or such transmission of energy by radio, and to the licensing and regulating of all radio stations as hereinafter provided…”

That's from the 1934 Communications Act. You're sure nothing has changed legislatively about the FCC in 92 years?