Scientists have to list all the sources they use. And they quickly get called out for doing mistakes in that regard and suffer a loss of trust in their work.
What would happen if everything politicians say or write had to contain sources?
Speeches are prepared anyway, so you have to publish all the sources of your speech right after you held it. Saying things differently than in the source would be illegal.
I think it would be quite interesting, and a completely different way to do politics.
Look at politicized science for your answer. I know you were asking what would happen to politics and politicians if they followed the scientific process and followed the rules legitimately seeking the most correct answers, but what they’d do in real life is cherry pick data, use poor “n” and minimal data collection to steer the results how they want, then selectively interpret the data to support their narrative with clickbaity titles attached to the results. Just like industry funded science did for cigarettes or leaded gasoline.