To add to that, the original TV signal is high quality, and digitized. All of the original content is there in digital form. The digital signal is then pumped out on an RF broadband carrier (cable). The TIVO box reconstructs the digital signal. The output from the TIVO can be coax RF baseband, not broadband (lowest quality signal with embedded audio), composite (yellow wire for video, red/white for stereo audio), still pretty awful), 'S' video (a step up from composite, but still poor, audio on separate wiring), component (3 wires for video, 2 for audio), and HDMI (one cable v1.4, carries full HD 1080p compliant video, plus 5.1 audio, plus reverse controlling cabability, plus 3D protocols).
It makes all the sense in the world to provide the BEST signal quality you can to your TV.
Analogy: Alaskan Pipeline carrying hundreds of thousands of gallons a minute (cable). Somewhere on that pipeline you tap off the flow into a straw (your RF cable to the TV) to fill a tanker truck (your TV). Replace the straw with a 6" hose.