Why can't I do a loop cut after Merging these vertices? So, I have this geometry: 



It was created from this geometry:  



by simple scaling to zero about the 3D cursor, which had been moved to selection, but selecting the topmost and bottom-most vertices.  Now, as you may already know, this isn't nearly as connected as it looks; the fact all these edges are intersecting doesn't mean there is a vertex there to unite them. Let's try to fix that with:
 

But no dice, it becomes impossible to do loop cuts.  This:


becomes impossible after the merge.  Could someone be so kind as to tell me why?  Is there a way around this problem, or should I just do my loop cuts first?  

Doing the loop cuts first seems to work, but I'd like to understand the nature of the problem.
The fundamental problem is that loop cuts work on quads; they are interrupted by triangles, and faces with more than four edges. You can do the loop cut before you merge all the points together, because Blender treats each face as a quad, that is a face defined by four vertices, where one edge has zero length because two of the vertices have the same co-ordinages. The merge makes all of quads triangles, because two of the vertices on each face are combined into one. Similarly, if all of the three or more vertices of a face have the same co-ordinates, you still have a face, even though the face has zero area.

With respect to achieving an effect similar to the loop cuts, switch to edge select mode, select all the radial edges (but not the circumferential ones), and use the subdivide tool, setting the number of subdivisions to the desired value. Another way is to use the knife tool. Yet another way is to extrude the vertices around the circumference, and scale them almost to the center, then extrude again, and scale the second set of vertices to the center point, and merge the center vertices together. Now you can still make your loop cuts in the quad faces created by the first extrusion, and you can select the edge loop created by the second extrusion and use the edge slide to position it where you want it.