Are there ways to automatically (no human testing) measure a $9 \times 9$ Sudoku puzzle's average hardness for a human to solve?
There are have been many such attempts. Most of them try to derive deduction rules which humans seem to use to solve Sudoku puzzles.

My money is on this approach:

Mária Ercsey-Ravasz &amp; Zoltán Toroczkai (2012), The Chaos within Sudoku, Scientific Reports 2:75, Nature.

The idea is based around the notion of transient chaos in a dynamical system.

In dynamical systems, when you change the state, there is often a period of time where a transient signal dominates, before the system manages to get itself into a steady state solution. If the dynamical system is nonlinear, the transient may be chaotic.

This is different from a chaotic attractor (e.g. the Lorentz attractor), in that the steady state that the system is eventually drawn to isn't chaotic, just the transients.

The time that the system takes for the transients to die down is called the escape rate. What the authors found is that if you turn the puzzle into a SAT problem, and then turn that into an equivalent dynamical system (where the attractor of the dynamical system is the solution to the SAT problem), the escape rate of the system correlates quite well with how Sudoku puzzles are rated for hardness.

The interesting thing about this approach is that it's independent of Sudoku. Any puzzle which can be turned into a SAT problem fairly directly (the SAT problem and the puzzle can't diverge too far) can be analysed in the same way.