Trust is the conduit for influence; it's the medium through which ideas travel.
It's not uncommon for people to overvalue the importance of demonstrating their competence and power, often at the expense of demonstrating their warmth.
Being a comfortable public speaker, which involves easily being able to go off-script, strongly signals competence.
I sometimes work with a communications and media training firm called KNP Communications. It's nice to bring the research to the practitioners; I learn a lot watching how they put it into practice, and I know they like to be on top of what's happening on the research front.
Politicians are very experienced - maybe too experienced - at using body language to signal power and competence. But what these politicians are much more likely to struggle with, or just neglect to do altogether, is communicate warmth and trustworthiness.