I feel the same way about inverting a binary tree, but a lot of people act like it's an arduous request. I am guessing it's because they've never read the description of what inverting a binary tree is, but maybe people are just that bad at recursion.

You can go your entire career without recursing, or using a tree data structure in its raw form (i.e. you only use it as part of a library)

Right. For the first many decades of computing, recursion was just always the wrong answer for a production software system. (Feel free to provide a counter-example, but please begin with an explanation of how the size of a call stack frame is determined and how exceeding the base allocation is handled on this platform).

So what tree-traversal/quicksort problems tend to measure is how long it's been since you last did CS class homework problems.