Making Rounded Diamonds?

Gareth,

Just to go vintage with the whole solution, based on the Common Tangents script answer by Kurt with addition of another script from the same collection and rounding up in a way that has been with us for decades (see comments below):

1 ) Starting with the paths,

2 ) Select both, then apply Common Tangents,

3 ) Select all, then apply Join Reasonably,

4 ) Apply Pathfinder>Unite (different wording over time).

Click to get closer, Click again to get closer still

You can find the scripts in the free set of scripts by 佐藤浩之 (Satō Hiroyuki), either here or here,

https://github.com/shspage/illustrator-scripts/tree/master/etc

https://shanfan.github.io/Illustrator-Scripts-Archive/

Comments:

The description and images are based on the two closed paths in 1). If there are open paths or two smaller circles instead of the long path with rounded ends, it becomes a bit more complicated, as follows:

2 ) If there are two small circles together with the large one, there will be superfluous Common Tangents which should be removed so only the four relevant ones remain (quite easy).

3 ) If there are any other open paths than the four Common Tangents, the Join Reasonably will lead to a mess instead of the orderly creation of the octagon including them,