I can offer a few comments on your project, although without solving your specific problem.
I believe my outlook is rather different from yours. Historically, statistical mechanics arose from macroscopic thermodynamics; you cannot divorce the two. We can take <PERSON> and <PERSON> as the creators of statistical mechanics, with a hat tip to <PERSON>.
For simplicity, I focus first on <PERSON>, who was primarily a macroscopic, classical thermodynamicist. In his day, most thermodynamicists were adamant that their science was purely macroscopic, and had nothing to say concerning microscopic reality (presumably atomic and/or molecular.) <PERSON> set out to create a mathematical structure which would start from the Hamiltonian mechanics of molecules, and end up with a thermodynamics whose variables (energy, entropy, etc.) were all expressible in terms of molecular parameters. The success of his effort is manifest in the fact that he worked before the discovery of quantum mechanics, and yet his conceptual structure is still what we use today.
Gibbs' work is still in print, but difficult to read. I recommend <PERSON> (Statistical Mechanics, now available as a Dover reprint) and <PERSON> (Statistical Physics, Vol. 682
1.) You will have to get beyond <PERSON> if you wish to make progress here; and I would be wary of trying to build your logical structure upon his statement of fundamental principles: for example, two states of equal energy will have different statistical weights if one is degenerate and the other not.
<PERSON>'s aim was similar to that of <PERSON>: i.e. to connect macroscopic thermodynamics with microscopic molecular mechanics. He was roundly attacked for this, and it is likely these attacks contributed to his suicide. Unfortunately, he did not see <PERSON>'s early papers on Brownian motion (which had already appeared) and which provided the first definitive verification --i.e. with respect to a specific physical system-- that statistical mechanics correctly describes thermodynamics.
The things you wish to avoid --Hamiltonian mechanics, ensembles, phase spaces, etc.-- are not easy to avoid, and attempting to avoid them will likely lead you into much frustration. The <PERSON> equation is not a necessary starting point: if you begin with a Hamiltonian, you get a statistical mechanics which is readily adaptable to either classical or quantum-mechanical usage.
I know my answer is not what you are looking for; but I hope it is some small help in spite of that. 682