Seems easy to strip off or not add, and then you'd have a video with unknown provenance which the echo chamber will eat hook, line, and sinker. Wouldn't you?
The best option imo is for independent organizations that people trust to handle fact checking, and have such organizations fact check each other. I also think it's completely appropriate for the government to fact check news agencies as well, provided they don't attempt to shut down orgs with a poor track record.
We should still educate people better, but there's still going to be a ton of people falling for nonsense like this.