an axiomatic framework for clustering by jon kleinberg may have a problem?
I do not want to attempt a general definition of "unnatural", but I find your argument about the inherent incompatibility of the scale-invariance axiom and the richness axiom as defined in Section 2 of Kleinberg's NIPS paper.

The impossibility result in Section 3 of his paper is stronger, however, and can directly address your concern.  


  Theorem 3.1 If a clustering function f satisfies Scale-Invariance and Consistency,
  then Range(f) is an antichain.


So you can consider weakening the richness axiom, e.g. by asking for all partitions with at least 2 parts, and as long as your set of partitions is rich enough to not be an anti-chain, the impossibility theorem goes through.

Hopcroft and Kannan have come up with a "natural" alternative to the richness axiom for which clustering is possible.  It is buried in their book Computer Science Theory for the Information Age, and described in a lovely teaser on the Gödel’s Lost Letter blog.