In this video, the instructor asks which arbitrary value of $\hat C$ produces as much simplification as possible to the system $\begin{cases}\frac{dN^*}{dt^*} = K_{_{max}}\tau \frac{C^*}{\frac{K_n}{\hat C} + C^*} N^* - \frac{F}{V} \tau N^* \\ \frac{dC^*}{dt^*} = -\frac{\alpha K_{_{max}} \hat N \tau}{\hat C} \frac{C^*}{\frac{K_n}{\hat C} + C^*} N^* - \frac{F}{V} \tau C^* + \frac{F}{V} \frac{\tau}{\hat C} C_0\end{cases}$. The choice proceeded with was $K_n$, but my initial reaction was to make $\hat C$ a very large number, such as a googolplex, or $\lim_{n -> \infty} n$. For any fixed values of the other parameters, one could find a value of $\hat C$ large enough to make a good portion of the complication in the system practically disappear. This would make solutions to the system slightly imprecise, but in applied problems such as the one being considered, this should be much less error-inducing than measurements of any empirical data used as parameters.
This might be cheating, because it makes hard problems easier than it feels like they should be. Nonetheless, if the constants are truly free, then they should be able to be made arbitrarily close to infinity. This might change the nature of analytic solutions, but numeric solutions at least should remain accurate.
Is this a valid trick to nondimensionalize systems? If not, why doesn't it work, and is there at least some weaker relationship than nondimensionalization that the resulting system would have with the original system?
So it is not an option to simply set the scale parameter equal to infinity and discard terms that involve division by it. This is "cheating" because it doesn't give the nondimensional variables a chance to "respond" to your choice.
However, it is perfectly fine to choose it to be huge or small. But it is not nearly as magical as you think it is, and in fact can be actively counterproductive.
Here for instance you can identify $K_n/\hat{C}=\varepsilon$ as a small parameter, so that $\hat{C}=\varepsilon^{-1} K_n$. Then you get
$$\frac{dN^*}{dt}=K_{max} \tau \frac{C^*}{\varepsilon + C^*} N^* - \frac{F}{V} \tau N^* \\ \frac{dC^*}{dt}=-\frac{\alpha K_{max} \hat{N} \tau \varepsilon}{K_n} \frac{C^*}{\varepsilon + C^*} N^* - \frac{F}{V} \tau C^* + \frac{F}{V} \frac{\tau \varepsilon}{K_n} C_0.$$
What goes awry is that if $C$ (or whatever the dimensional variable being scaled by $\hat{C}$ is called) isn't actually huge compared to various things with the same units as it ($K_n$ especially), then expressions like $\varepsilon + C^*$ lead us astray in the forthcoming analysis of the nondimensional problem. This is because we are tricked into thinking $\varepsilon \ll C^*$ (and thus are tempted to start expanding in powers of $\varepsilon/C^*$), but this isn't actually the case.
On the other hand, maybe $C$ is actually huge relative to $K_n$. If this occurs then this method allows you to study the nondimensional problem as a regular perturbation of a simpler nondimensional problem.