It's not like someone said "Ah ha, torque! What should the definition of torque be?" That doesn't make sense; you don't think up a term and then try to assign a definition.

Instead, it was found that this thing $\mathbf r\times\mathbf F$ turned out to be really useful in explaining physical phenomena. Particularly rotational dynamics of systems. So it got its own special definition.

If you want to define $r^2F$ or $r^2F^2$ as something, you can go right ahead. No one will stop you. The goal would take be to show that your new definition has physical significance.

P.S. I think the answers that have "Define torque as the time-derivative of angular momentum" or something like it is showing the physical significance of $\mathbf r\times\mathbf F$ by linking it to its physical significance (assuming you think angular momentum and its rate of change is significant). But I think they miss the point of showing that you can't derive or prove definitions and that questions like "why is **insert physics term here** not equal to **insert modified definition here** instead?" misunderstand why something would be defined in physics in the first place.

What you are doing here is not actually asking why is torque defined as $\mathbf r\times\mathbf F$, but rather (possibly unaware to yourself) you have some *other* notion of what torque is/should be, and you want to know why we can go from what you are thinking of to $\mathbf r\times\mathbf F$. Of course, we would need to know what your actual starting point is that you desire in order to get there. Other answers assume rate of change of angular momentum, or use virtual work and equilibrium, but that isn't necessarily what you might be thinking of. The variety of other answers here shows how flawed the question (not the post though) is.