Specifically, I've found through my own user testing that certain "fancy animations" are important to helping users build a correct mental model of the current workflow—taking them away makes the user take longer to learn the application. For example, the window-minimization animation on macOS: it shows you where to find the window you just minimized, by showing you the window "going into" the Dock rather than just disappearing. Or, for another example, disclosure indicators (triangle buttons), where in most modern app frameworks the results sort of "unfold" into place below the indicator, rather than just appearing. Having the results just flash into place, especially if it causes the document (and thus the scrollbar) to change size, tends to disorient the user.

Having as little useless decoration is commendable, but I'd advocate for user testing on mockup versions of the software that contain some minimal animation, vs. those that don't. (Don't trust the user's opinions on which is better; just watch which version they get used to faster. Basically, do the same thing that game developers do when testing interaction mechanics: find the optimal "communication path" between the user and the software.)