Let's say there is a hot battle for first. You want to follow the fight closely and seamlessly (no transitions to other shots). So far I've only seen this in videogames; drone cams which I was delighted to read about elsewhere here are a great cost-effective step toward this but they still can't maintain close proximity or keep up with the cars.
For reliability it seems least parts is the best solution, so my initial thought was a "maglev" arrangement for the bracket holding the camera and charged guardrails (I know how this sounds! But only enough current to levitate a camera and bracket). Other solutions would be a pulley system but I think would be more expensive to implement. And ofcourse, when cars do crash into it, you are destroying a guardrail with some wiring through it, not some expensive mechanical system (a la greyhounds chasing toy rabbit-system they use in those races)
This could be implemented on any track with rich owners/on a healthy motorsports rotation. Imagine the benefits!! Imagine having multiple cameras on this arrangement and you use the coloured buttons on your remote to follow whatever part of the action you desire!!! They could also be semi-automatic by having simple sensors that follow the car but ultimately will be fine-adjusted by the cameraman (perhaps to focus on brake disk glowing, or damage, etc etc ETC!!! There is so much potential for such a system, and unparalleled footage!
Please feel free to bring me back down to Earth below, I'm sure people thought of this as soon as the first racing games with replays came out but apart from financial reasons (which could be offset by Ecclestone and people in similar positions in other areas) I am not seeing any drawbacks, especially in conjunction with normal filming!
PS - Maglev arrangment will also give the best quality footage, free of vibrations etc, though the most plausible is BATC42's idea of a more "traditional" self-propelled camera