A pointy triangle could be idealized as an edge, the resulting circle will always be bigger than a bounding box.

Indeed. Any particularly flat (in either Cartesian direction) ploygon is going to be better suited to a bounding box. Whether these are plentiful or rare in the data set is only something which the OP can answer.

You might now claim that something like a regular octagon is better represented by a circle (probably).

I'd claim that not only is any regular polygon with n>4 better represented by a circle, the circle is trivially easy to compute. If the data set were all regular polygons (and I'm not assuming for one minute that they are in this case) then circles would be the obvious choice.

A little domain knowledge here would help to inform the choice of bounding box or circle. Horses for courses.