It's the "non-argumentative ways of conveying a point of view" that I find uncomfortable. How do you properly respond to them?

The same way I properly respond to all other ways of conveying a point of view: the most effective way I know that isn't itself unethical. If my audience is more moved by poetry than prose, arguing in prose when others are using poetry to argue wrong positions is not praiseworthy.

Of course, if I've spent decades learning how to argue in prose and/or am naturally skilled at it, whereas I'm not skilled at arguing in poetry, that means choosing to compete in an area where I'm not confident.

And you're absolutely right: that is uncomfortable. Sometimes, the right thing to do happens to be uncomfortable. It sucks, but there it is.

All of that said, it is important to clearly distinguish in my own head between the goal of being compelling and the goal of being correct. They are distinct, and largely orthogonal. But they are both important.