Earlier quoted context omitted.
Maybe I'm wrong, I'm a very causal user, but I think the only way to do it is to create a shape that covers the part you want to keep, and then do object -> set -> clip or something like that; it's not exceedingly awful, but it's not as smooth as clicking "crop" and dragging borders.
But the upside is that you can crop to any shape, not just rectangles. Also, there are not 2 workflows, one for rectangles and one for arbitrary shapes, which is a good thing.
If I were approaching the problem fresh, though, I would look into the presentation of the modal functions as the root of the issue. The source of a lot of complexity in image editing software is confusion over layers, masks, groups, and selections - they have some conceptual overlap and can be applied modally(e.g. shape as rendering element OR mask OR selection). If the UI buries the difference it just adds to the confusion.