Former journalist here. I would argue that it's a shared-responsibility model. We, the public, are at least partly (and I would argue mostly) responsible for developing the media literacy that helps us end up with the right understanding, rather than requiring media outlets to publish general disclaimers and PSAs.
When I was in high school, I took a one-semester media literacy course where we examined topics like reputable sources, bias, sensationalism, moderating one's consumption, why watchdog reporting is so important but often goes unnoticed, etc. I would love to see more high schools offer this.