Turn ON smart guides and snap to point. Turn OFF all other snap options.
Let Illustrator show the centers of the objects. This can be turned OFF, so it's possible you have done it and you must turn it back.
Draw a circle. Select it with the normal black arrow.
Hold the Ctrl key (this turns temporarily the black arrow to white) and drag the centerpoint of the circle on the wanted anchor point. It snaps.
This does not work if you have clicked anything else on the circle with the white arrow after selecting but before the dragging.
The target anchor point must be a real one, not one which becomes real after a still alive and editable effect is expanded nor an effect adjustment handle.
If it happens that you are not missing the thing you asked, but how to snap your circle or shape center to something else than to an existing anchor point, this method is not a solution.
But I have still something. Sometimes people want to find the the centerpoint of a circle. They unfortunately have only an arc of it, so the shape center is useless.
See image A in the next cartoon
Let's assume the red curve segment (the whole red part, nothing less) is a circular arc. The black curves are not tangential lines of the arc. If they were, the case would be a little simpler.
To find the centr draw 2 chord lines (green and magenta in image B). Be sure they snap to the arc at both ends.
In image C both chord lines are rotated 90 degrees. The pivot points are the default ones (=midpoints). The wanted center is the crossing.
In image D a circle is drawn to the chord intersection. To get it draw an ellipse holding Shift+Alt at the same time. Or move an already drawn circle to the crossing like told above. It snaps also to the intersection
If the black curves in Image A happened to be straight tangential extension lines of the arc, you simply draw their duplicates and rotate them both plus and minus 90 degrees, The rotation pivot points must be the arc ends. It's sketched in the next cartoon.
If you browse older questions you probably find this and numerous other tricks where elementary geometry is applied to a drawing problem. Tricks are sometimes cool, but often eyeballing with high zoom in is much faster way to draw accurately enough for a nice looking image. Engineering needs something different and Illustrator is not the best tool for it.
The center of the circle will snap to the square anchor points in your picture (but not to the corner widget itself).
First make sure View> Snap to Point is checked on in the Menu Bar.
Also, make sure Show Center is on in the Attributes Panel- Window> Attributes
Then use the White Arrow (Direct Selection Tool)- it seems to me to have much better snapping capability than the Black Arrow (Selection Tool). Option/Alt click on the circle to select the entire circle shape (simply clicking the shape with the white arrow will only select the path or anchor you directly click on and not the entire shape). It is important to drag your selected shape from the anchor which you want to snap to the rounded corner anchor point. This also works when dragging from the center point.
If you are still unable to snap then go to Preferences (Settings now an mac) and under Selection & Anchor Display make sure the Snap to Point item is checked on and maybe even increase the value to a higher number. In the example mine is set to 15 px.