Firstly, people already know politicians lie. Nobody is going to care about your video in the first place.
Secondly, your analogy is wrong. The new hypothetical you present is equivalent to the old model where you are all giving verbal accounts and his staffers lied. If you are going to compare to an old-tech scenario where you one-up them with a more detailed recording than they have, you need to do the same with new technology to keep the comparison accurate.
Thirdly, why aren't you the liar?
Fourthly, you can always go one level up. Show the footage of them constructing doctored footage.
This only works in large events. What if you film a politician lying at a small event, but his image team makes 20 fake videos showing the event in a different light? I guess you're the liar then.