Problem connecting to paths seamlessly

ep,

A corresponding mad but accurate way if you wish to keep the height difference between the circles, Smart Guides and Line Segment Tool being your friends, one more step:

1) With the Line Segment Tool ShiftClickDrag between the left and right Anchor Points, then lock it,

2) ClickDrag the small circle by its right Anchpr Point to snap to the left Anchor Point of the large circle,

3) Rotate both circles round the centre of the large one by the centre of the small circle to snap to the line from 1), then ShiftClickDrag with the Line Segment Tool from the centre past the top, then lock it,

4) With the Scissors Tool cut off the small bit of the small circle.

Click/RightClick to get closer and Click again to get closer still

ep,

A mad but accurate way if you wish to keep the angle between the circles, Smart Guides and Line Segment Tool being your friends:

1) With the Line Segment Tool ClickDrag between the centres, then lock it,

2) With the Scissors Tool cut off the small bit of the small circle,

3) ClickDrag by the new end Anchor Point along the line until snapping to the intersection with the large circle.

Click/RightClick to get closer and Click again to get closer still

Turn off the grid. I am afraid that you need to nudge that manually with a very small keyboard increment.

Or use intersecting guides and turn on Smart Guides.

But even if you align the anchor with the path, you need to delete parts and change the curve to get a smooth transition.

In your preferences, try unchecking Use Preview Bounds. That should allow you to align your shapes based on the the outlines instead of the dimensions of the strokes.