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.
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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.