If (and that’s a big if) animation is used in moderation only when it actually communicates something and isn’t an active impedence (as demonstrated in the linked post), I think it has a significant effect for users. It’s just not the effect that many might expect.
Meaningful, unintrusive animations are one of the myriad puzzle pieces that come together to form a positive impression. They’re a sizeable chunk of that last 20% that separates “good” and “excellent” in users’ minds. They’re not strictly necessary, but between two equally good competitors they’ll help one pull ahead of the other, because users come away with a stronger impression of “solidness”. It’s not unlike how people tend to consider heft and resistance to flexing as markers of higher quality in physical products.
The problem is that since a decade or so ago, UI design as a whole has veered heavily in the direction of vibes, slideshow wow factor, and “branding value” (I felt a pang of nausea just writing that) and away from the volumes of well-researched best practices, and regard for good use of animation has been lost along with it. We’re well overdue for a correction that pushes UI design back in the direction of practical usability and away from Dribbble appeal.