Indirect complement or extraposed subject?

Is the content clause ('that she saw him commit the crime') in the following sentence an extraposed subject or an indirect complement* licensed by 'so'?

It just so happens that she saw him commit the crime.

*See, for example, The Cambridge Grammar of the English Language, p. 443.

Solution 1:

It just (so) happens [that she saw him commit the crime].

It can't be extraposition since the content clause cannot occupy the subject position: we can't say *"That she saw him commit the crime just so happens".

And I wouldn't say that the content clause is an indirect complement licensed by "so". "So" is optional here; dropping it would not result in a loss of grammaticality or a real change of meaning.

Rather, the content clause seems to be a straightforward complement of "happens", the item that licenses it.