What is the significance of the sending forth (ite missa est, from which the Mass gets its colloquial name) if not to go and do the work of God? That's what the announcements should be about--parish life and what is going on for people to take part in. Mass is your only time of captive audience; many people do not read the bulletin; many more don't read all of it, even if they intend to. Announcements are to call attention to the most important thing, report on important things that are difficult to convey easily in a busy bulletin, and invite people to do God's work with your fellow parishioners.
As such, I think their proper place is either directly before or directly after the sending forth. I believe that is included in the liturgical rubrics, if my memory serves from past discussions on this topic.
I would issue this question (challenge, perhaps?) to those who don't want any announcements included, or who find them a distraction. Do you merely internalize your faith? Is the Mass just an internal worship? Are we not suppose to take our worship (and the Eucharist we just received into our bodily temples) out into the world in living action?
That's what the announcements and the life of the parish should be about. I would go so far as to suggest that there is not just a missed opportunity but a great gap, a disconnect between faith and our faith alive through works, if announcements are not connected to the Mass. Not connecting them is like saying our prayer and worship is separate from anything else we do, rather than uniting the two and letting the Sacrifice and worship we just committed give life to our parish and our work.
Essentially what I'm saying is that disconnecting announcements (about parish life and ministry) from Mass is like glaringly highlighting that "faith without works is dead..."