Sorry, my phrasing appears to have distracted from the intent of my question. I just used the rectangle as an example, because I didn't know how else to phrase it. I want to use the circle to cut the corner off of any object, not just make rounded rectangles. (I'm aware of the rounded rect tool.)
Put another way, the circle in that picture is surrounded on all sides by the larger object. I only want to cut off the outside part (the corner). Is there an easy way to do that?
This is the process I tried to illustrate above. Hopefully it will be more clear. I'm wondering if this can be done with just the boolean functions, instead of deleting individual nodes.
I want to remove the corner at the arrow.

Add two nodes on the rectangle, as close as possible to where the circle meets it.

Delete the corner, and make a straight line. (I moved the circle underneath for this shot to make this clearer)

Do a union, and it's all done.

This is pretty easy with a circle, but imagine if you were using a star or irregular shape. Is there an easier way?