I think it's one of those things where you don't "notice" it, but where it nevertheless has an impact. Sort of like someone might not "notice" the fact that there's more butter or salt in restaurant food, but it's subjectively better than the same meal they cooked at home.

For a more directly relevant example, companies frequently A/B test changes to a UI to see which ones people like better. The specifics of those changes would be pretty marginal if you didn't know what it looked like before (like if you're a new user, you wouldn't notice if the notification was red versus purple, or what the wording in the menu is). Despite this, there are some sites that just "feel" better in a way that you can't really describe.

All of this is a long-winded way of saying that I can't tell if I'm looking at Arial, Helvetica, or this Nebula Sans font unless they were side by side (and even then I'd just be saying they're different, not identifying them by name). But I think the site would feel a lot less modern if it were written in Times New Roman. I think you'd notice if it were too hard to read when small, and I think if it looked "bad," you'd at least subconsciously notice that.

You’re ignoring half of the argument. I have given some examples of branding and some examples of real use. “Legibility, and understanding” aren’t branding. “Dyslexia” isn’t branding. Typography is a branch of graphic design with a lot of study behind it. Just like subtle uses of colour and element positioning can change how you interact with, perceive, and feel about an interface, so can typefaces.

Again, just because you’re unable to notice how exactly you’re being affected does not mean you aren’t. You also don’t notice all the ways you’re affected by advertising, but they still work on you.

Yes, of course not every single subtle change to a font makes a huge impact. Just like a single subtle change to a colour’s hue doesn’t. But pronounced changes do, even when you’re unable to put your finger on it.