Could temperature have been defined as $-\partial S/\partial U$?

For a mathematician, the answer is that the customary calibration of temperature---Kelvin's absolute temperature---has enormous advantages over all its rivals (including the one proposed in the above query): it is essentially the only one that leads to the mapping from the pressure-volume plane into the temperature-entropy plane described by the equations of state being area-preserving (recall that this area in both planes can be interpreted as energy). Analytically, this means that if the equations of state have the form $T=f(p,V), S=g(p,V)$ then $f_1g_2-f_2g_1 = 1$ (subscripts denote partial derivatives) and this is equivalent to the Maxwell relations.

We quote from Kelvin's original article: "The characteristic property of the scale which I now propose is, that all degress have the same value; that is, that a unit of heat descending from a body A at the temperature $T$ degrees of this scale, to a body B at the temperature $(T-1)$ degrees would give out the same mechanical effect, whatever the number $T$. This may justly be termed an absolute scale."

One of the central incidents in the history of physics is surely the search for the true scale of temperature (Maxwell's treatise on Heat begins with a lucid discussion of the problems involved) and its solution (the practical one due to Regnault and the theoretical one due to Kelvin---it is no concidence that the latter had worked in the former's laboratory) must be one of the most significant scientific achievements of the 19th century.

If one takes $U$ as the dependent variable and $N, V,$ and $S$ as the independent variables, then one has $U = U(S,V,N)$ with the total derivative of U equal to $$\rm dU = (\partial U/\partial S)_{V,N} \; \rm dS + (\partial U/\partial V)_{S,N} \;\rm dV + (\partial U/\partial N)_{S,V}\;\rm dN.$$ Each of these partial derivatives has a "simple form", with $T = \partial U/\partial S$, $\,-p = \partial U / \partial V$ and $\mu = \partial U/\partial N$. The condition for equilibrium between two systems open to thermal transfer is that the two $T$'s be equal, the condition for two systems open to pressure change is that the two $p$'s be equal, and the same for exchange of particles with the two chemical potentials $\mu$ be the same.

These intensive variables are fundamental for thermodynamics, but could have been defined in any reasonable way, such as $1/T$ ($1/k T$ might have been better) or whatever. So the short of it is that while the derivatives are all important, their actual definition is somewhat arbitrary.