The definition of speed (please, let me call it velocity hereinafter) is not random at all.

It seems you understand that it must depend on the distance $d$ and the time $t$, so I'll skip to the next stage.

Evidently (for a constant $t$) velocity increases if $d$ does; and (for a constant space) $v$ decreases if $t$ rises. That constrains the ways we can define it. For example, your example of $d+t$ is authomatically discarded. You could say $d-t$, that satisfies the growing conditions.

Then we apply the reasoning in the limit case. For a 0 distance, velocity must be 0 independently of time (unless time is 0 too), that discards any sums. If the time to reach the space is infinite, the velocity must be 0. That's forcing $t$ to be a denominator.

So we deduce it's a fraction, but how can we sure there are not powers of those quantities? We impose the linearity of space. It doesn't make sense that the velocity is different if you pass from 50 to 60, or from 70 to 80 in the same time. If all points in space are equivalent, there cannot be distinctiosn like these, so using the numerator $\Delta d$ guarantees that all points in space are equivalent. IF it were $\Delta d^2$ the result would be different from 70 to 80 and from 50 to 60, for example. That's againts the obvious principle that we can set the origin where we want (we must eb able to measure from the point we choose, as we do everyday with a simple ruler, placing it where we want). The same reasoning applies to time.

So they must be a fraction, and there cannot be other powers than 1. The only possible difference is a constant factor

$s=k \frac{\Delta d}{\Delta t}$

And this is what speed (or velocity) is, after all. The constant is actually the unit factor. It depends on what units you are using. I hope this is useful to you.

The definition of speed (please, let me call it velocity hereinafter) is not random at all.

It seems you understand that it must depend on the distance $d$ and the time $t$, so I'll skip to the next stage.

Evidently (for a constant $t$) velocity increases if $d$ does; and (for a constant space) $v$ decreases if $t$ rises. That constrains the ways we can define it. For example, your example of $d+t$ is authomatically discarded. You could say $d-t$, that satisfies the growing conditions.

Then we apply the reasoning in the limit case. For a 0 distance, velocity must be 0 independently of time (unless time is 0 too), that discards any sums. If the time to reach the space is infinite, the velocity must be 0. That's forcing $t$ to be a denominator.

So we deduce it's a fraction, but how can we sure there are not powers of those quantities? We impose the linearity of space. It doesn't make sense that the velocity is different if you pass from 50 to 60, or from 70 to 80 in the same time. If all points in space are equivalent, there cannot be distinctions like these, so using the numerator $\Delta d$ guarantees that all points in space are equivalent. If it were $\Delta d^2$ the result would be different from 70 to 80 and from 50 to 60, for example. That's againts the obvious principle that we can set the origin where we want (we must be able to measure from the point we choose, as we do everyday with a simple ruler, placing it where we want). The same reasoning applies to time.

So they must be a fraction, and there cannot be other powers than 1. The only possible difference is a constant factor

$s=k \frac{\Delta d}{\Delta t}$

And this is what speed (or velocity) is, after all. The constant is actually the unit factor. It depends on what units you are using. I hope this is useful to you.