Pull does not solve the problem. The attack here is the metadata "what was the origin" and "what is the destination", and pulling data does not solve that.

There was some discussion on techniques to get around that in this HN post about a messenger type product: https://news.ycombinator.com/item?id=37105477

Sorry, I thought the problem was that the pushes were going through Apple's and Google's systems, which direct pulling would avoid.

How about messaging via wide-spectrum dead-drop hopping? (redundant array of inexpensive drops?)