I teach a Mathematical-Statistics course and I have chosen this example (rather randomly, among others) to demonstrate to students how does one find the method-of-moment estimator (no problem here, including asymptotic distribution) and the ML estimator - no problem finding it numerically when using a computer, but so far impossible to prove more than its consistency.

Yes, that's correct. But the problem is that the limit (using L'Hospital) requires higher and higher derivatives (something like $2k$ for a $kxk$ problem, depending on how you go about it), which is not feasible beyond $k=5$, whereas Carlo's formula can be easily applied to $k$ in the hundreds or more.

Not that there was ever any doubt, but your second-derivative formula verifies for up to 5 by 5 matrices (as far as one can get by brute force - after that, it would be algebraic nightmare. This is to emphasize the formula's importance: one cannot bypass it by taking a numerical limit for even relatively small matrices; this becomes an extremely ill-conditioned task).

Sorry. my stupid mistake: I was talking about the (elementwise) derivative of the corresponding INVERSE (now corrected)!

# of spanning trees equals to $det(C.C^t)$ - any such formula for # of acyclic cotrees?