With the measure tool active, I am allowed to select vertices/faces/edges.
This should not be allowed, unless the measure tool actually places a measurment on that selection!
So, my desired outcome would be:
- Clicking on a vertex starts a distance measurment being drawn. Clicking again completes it (snapping support should be allowed here).
- Clicking on an edge puts a complete distance measurment on that edge.
- Clicking on a face puts an area measurment on that face. Multiple adjacent face selections should increase the total measured area.
The last point is probably one hell of a feature request, but the first two sounds reasonable, I hope. 
EDIT: Just discovered that point 2 and 3 is in the overlays setting already, but I think they also belong in the measure tool.