Why is conductivity defined as the inverse of resistivity?

In my experience this comes from resistance and conductance in electrical engineering and circuit theory. If you use the loop current analysis method on a circuit of resistors and sources then you get a matrix of linear equations whose coefficients are resistances. If you use the node voltage method on the same circuit you get a matrix whose coefficients are inverse resistances.

So the inverse of resistance shows up very often quite naturally in circuit equations, rather than the negative of resistance or the inverse of resistance squared. Because it shows up naturally it makes sense to give the inverse of resistance a name.

Usually when you run into some quantity that is defined and you are unsure why, that quantity first simply showed up in some important formula. So people needed a way to discuss that part of that formula, and so they gave it a name. But the quantity showed up on its own in the math first and was given a name later.

The usual definition matches up with calculations with parallel resistors. For example, the total resistance of two parallel resistors is $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}.$$ If the resistors have the same size and shape, then this can be written in terms of their resistivity like so: $$\frac{1}{R} = \frac{A}{L}\left(\frac{1}{\rho_1} + \frac{1}{\rho_2}\right),$$ where $A$ is the cross-sectional area and $L$ is the length of the resistor. In terms of conductance and conductivity, we can write this equation as $$G = \frac{A}{L}\left(\sigma_1 + \sigma_2\right).$$ In fact, just looking at conductance, the equation for a set of parallel resistors is much more intuitive: $$G = \sum_i G_i$$ as opposed to the usual $$\frac{1}{R} = \sum_i \frac{1}{R_i}.$$ The total conductance of a set of parallel resistors is equal to the sum of the conductance of all the resistors. This nicely parallels the case of resistors in series: $$R = \sum_i R_i$$ where the total resistance is the sum of the resistances of each resistor.

Using conductance and conductivity can be useful when trying to calculate the total resistance of a material whose resistivity varies across its geometry. See this question and this answer for an example.

The Ohm's law for a conductive material can be expressed locally as a linear relationship between the current density $\boldsymbol{J}$ and the electric field $\boldsymbol{E}$. For an isotropic material, this relationship can take either of the two equivalent forms: $\boldsymbol{J} = \sigma \boldsymbol{E}$ or $\boldsymbol{E} = \rho \boldsymbol{J}$. For these two forms to be really equivalent for a specific material, the relation $\sigma = 1/\rho$ should hold.

In the case of an anisotropic material, current density and electric field are no longer parallel and the above relationships take the forms $\boldsymbol{J} = \boldsymbol{\sigma} \boldsymbol{E}$ or $\boldsymbol{E} = \boldsymbol{\rho} \boldsymbol{J}$, where now $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ are matrices related by $\boldsymbol{\sigma} = \boldsymbol{\rho}^{-1}$.

Therefore, you cannot choose an arbitrary relationship between $\sigma$ and $\rho$, for otherwise you would lose one of the two equivalent relationships between the fields.