Screenshots go through the OS (for example, when you press the Print Screen button), or with third-party apps that use the OS APIs to record images or video. Sometimes this is also done via GPU drivers, like from AMD or Nvidia. These are on a software layer below the game, and thus the game has no control over this.
Beyond that, allowing players to take video of the game is great for marketing. As an indie, you are unlikely to get reviews from big sites like IGN, unless your game is already popular. So you will have to rely on small-time streamers or bloggers to take media of the game and talk about. Otherwise no one will ever hear about your game.