How can I make the corners/ends of an arc rounded?

clepperiousity,

As I (mis)understand it, you may:

1) Create the circle as a stroke/nofill path, with the Stroke set to Round Cap;

2) Create the rectangle on top of the circle, then Object>Path>Divide Objects below, then Direct Select one of the unwanted bits of the circle and press Delete twice, then Direct Select the unwanted stranght bits and press Delete.

Whether you use a stroke/nofill or fill/nostroke or stroke/fill rectangle, the circle will be cut at the spine of the rectangular path and the the roundingof the circle bits extends into the rectangle shape.

If you wish to avoid that,you may  use a fill/nostroke rectangle and Object>Path>Offset Path by half the circle Stroke Weight and dlete the original rectangle in 2).

Obviously, you may Object>Path>Outline Stroke afterwards if you wish/have a reason to have fill/nostroke paths.

Try this:

1. For each of the corners, zoom into the corner anchor point.
2. Select Direct Selection tool and click on the corner anchor point to select it.
   
3. The Corner Widget will appear (if you don't see the small circle,  View > Show Corner Widget).
4. Drag the widget inward to round the corner.
   
5. Repeat this for each of the corners until you get the desired corner rounding.