I passed on your question to John H. Conway. Here is his response: (NB. Everything following this line is from Conway and is written from his point of view. Of course, in the comments and elsewhere on the site, I am not Conway.)
I think it's wrong to focus on block designs in particular. This may not answer your question, but there are some interesting examples of theorems similar to Desargues's and Pappus's theorems. They aren't block designs, but they do have very nice symmetries.
I call these "presque partout propositions" (p.p.p. for short) from the French "almost all". This used to be used commonly instead of "almost everywhere" (so one would write "p.p." instead of "a.e."). The common theme of the propositions is that there is some underlying graph, where vertices represent some objects (say, lines or points) and the edges represent some relation (say, incidence). Then the theorems say that if you have all but one edge of a certain graph, then you have the last edge, too. Here are five such examples:
Graph: the Desargues graph = the bipartite double cover of the Petersen graph
Vertices: represent either points or lines
Statement: If you have ten points and ten lines that are incident in all of the ways that the Desargues graph indicates except one incidence, then you have the last incidence as well. This can be seen to be equivalent to the usual statement of Desargues's theorem.
Graph: the Pappus graph, a highly symmetric, bipartite, cubic graph on 18 vertices
Vertices: points or lines
Statement: Same as in Desargues's theorem.
"Right-angled hexagons theorem"
Graph: the Petersen graph itself
Vertices: lines in 3-space
Edges: the two lines intersect at right angles
Statement: Same as before, namely having all but one edge implies the existence of the latter. An equivalent version is the following: suppose you have a "right-angled hexagon" in 3-space, that is, six lines that cyclically meet at right angles. Suppose that they are otherwise in fairly generic position, e.g., opposite edges of the hexagon are skew lines. Then take these three pairs of opposite edges and draw their common perpendiculars (this is unique for skew lines). These three lines have a common perpendicular themselves.
Roger Penrose's "conic cube" theorem
Graph: the cube graph Q3
Vertices: conics in the plane
Edges: two conics that are doubly tangent
Statement: Same as before. Note that this theorem is not published anywhere.
Standard algebraic examples
Graph: this unfortunately isn't quite best seen as a graph
Statement: Conics that go through 8 common points go through a 9th common point. Quadric surfaces through 7 points go through an 8th (or whatever the right number is).
Anyway, I don't know of any more examples.
Also, I don't know what more theorems one could really have about coordinatization. I mean, after you have a field, what more could you want other than, say, its characteristic? (Incidentally, the best reference I know for the coordinatization theorems is H. F. Baker's book "Principles of Geometry".)
In any case, enjoy!