The OD describes apposite as mid 17th century: from French à propos '(with regard) to (this) purpose'. According to the Grammarist,
The loanword apropos comes from the French phrase à propos de, meaning with respect to. In English, apropos is conventionally used as a preposition meaning with regard to, and it’s also an adjective for pertinent or to the point.
This squares the circle for me. I might use the word as a preposition sometimes, but I would baulk at using it as an adjective (too pretentious) preferring words like 'right', or (more formally) 'appropriate'.
So, whereas these two work for me,
apropos of the initiative, it’s not going to stop the abuse
Isabel kept smiling apropos of nothing
this one does not
The ceremony concluded with the reading of an apropos poem
I prefer a less stilted adjective, in this context 'fitting' or, more formally, 'apposite'.