Rounded corners on crop tool
When crop tool is selected, add the live corners option to crop rounded corners or circles. Please and thank you.
There is an intended way to do it.
Once your image is cropped, you can do Object > Clipping Mask > Make and Ai will create and select a rectangular clipping mask for your image.
Then you have to convert it to a live shape, with Object > Shape > Convert to Shape.
Then you can round its corners in Transform panel, typing in a value in one of the Corner Radius fields (if they are linked) in the Rectangle Properties section.
All of these can be recorded as an action and applied later with a click in Actions panel or a hotkey.
Please test how it works for you and comment back.