The laws that you mention (e.g. Fourier law, Poiseuille law, Hooke's law), which involve a flux which increases as a material parameter increases, are interesting examples. Considering the original version of Ohm's Law was also phrased in terms of conductance, not resistance. This implies that in experimental physics, it's easier to watch something that's easy to change, and plot a linear relationship to something that changes, then it is to watch something that's difficult to change, and plot it against something that's easy to change.
It makes sense. You pull on a bar, and it grows. It's easy to change the "grow" by a set amount and stop, but difficult to get an exact pull force. You heat one end of the bar, and the heat spreads. It's easy to measure and you can stop when you reach a certain temperature. But you don't know how much heat you had to put in to make that happen.
It's also easy to apply different exact voltages to a wire, and watch the current change. Note the slope of that experiment would be the conductivity, not the resistance.
Why are you doing these experiments? So some day, you can make a bar that doesn't break when the load makes it grow. But, for now, you plot the thing that you changed v. the thing that also changed. And you call the slope "E", or "k", or "$\sigma$".
However, in engineering, it's different. While I use Hooke's law to figure out that bending my beam will lead to a displacement, inevitably I'm dividing by EI. Because I don't want my beam to deflect that much, I need a very large EI - but a small inverse would be better. I really wish I didn't have to divide - when I use a calculator it's easier to type $x_1 * x_2 * x_3$ v. $ x_1 / (x_2 * x_3) * x_4 $ - I'd rather we rate materials by 1/E, then by E, and the inverse of section modulus as well.
When calculating out fluid flow in a pipe, I'm always sizing my pumps not by dividing by the "flowability" of pipes, but adding up the friction factor resistance. I'm sizing my moving system to overcome this resistance, not to find out how my smooth pipes are helping the flow.
In heat transfer, I'm always dividing by the thermal conductivity. They even invented R-value on insulation so you could figure out the change in temperature between surfaces easier.
In electrical engineering, they didn't start with resistors - those are a manufactured device. So, they listed the value that mattered more to the engineers, and less to the physicist - the resistance. Unfortunately for Mechanical Engineers, we started with pipes and bars, so we keep with the same notation.