Previously on that same page it says that:
$$Cl_{tot}=\frac{X}{C_{SS}}$$
Where $Cl_{tot}$ is volume of plasma cleaned per unit of time, X is the amount of drug supplied per unit of time and $C_{SS}$ is the plasma contentration at the steady state. This formula is quite similar to the one in my question, although that concerns a single bolus dose, not a continuous supply of the drug.
The first thing I needed to do was to realize that this too describes a unit of time. More precisely, it describes the time between o and infinity. If $Cl_{tot}$ is interpreted as the volume of plasma cleared of that drug between 0 and infinity, the formulae are almost equivalent.
The only remaining stumbling block is whether or not $AUC_{0-\infty}$ can be said to be equal to $C_{SS}$. In a philosophical sense you can, since that "is" the total "sum of concentrations" which the body has gotten exposed to between 0 and infinity. We can make this even more similar to the formula for continuous drug administration by assuming that "sum of concentrations" could probably be spread out evenly over the entire timeline, giving us a mean steady state concentration (which would then hopefully yield the same result).
There might be some incorrect formalities somewhere but this is what made sense to me intuitively.
