Is a spectrophotometric method of measuring the rate of a complex reaction valid?
This is a perfectly valid method of determining rates of chemical reactions. It is obviously limited to those reactions where either the product or the initial reactant can be monitored easily using spectroscopy as well as reactions which are slow enough to get an absorbance measurement at set intervals. I have used this method multiple times in undergraduate chemistry labs and it works, and in principle it does not matter if there are intermediates.

The intermediates should not matter because the rate of a reaction is defined as the rate at which product is formed. That is, the rate of a first-order reaction is,
$$
\text{rate}=k_{tot}[A]
$$
If the reaction involves only one elementary step then the prefactor $k_{tot}=k_1$, the rate constant of the one step (assuming no backwards reaction). Note that $[A]$ can be the concentration of either a product or reactant, but you might need a stoichiometric coefficient in there depending on the reaction. On the other hand, if there are intermediate steps, the prefactor can become arbitrarily messy. Maybe there's an equilibrium between two intermediates so you have a term in there $k_2/k_{-2}$ and whatever else you can imagine. This doesn't matter though because your experiment measures the change in concentration at either end, and the slope is then the effective rate constant for the reaction which has all of the intermediate stuff lumped together.

In fact, this method you use is probably the most direct means of measuring a rate in the sense that a rate is defined as a change in concentration over time. You are measuring a change in absorbance over time, so there is only step (and one assumption) in going from your data straight to the reaction rate. That assumption is simply the validity of Beer's law, which is usually not a problem for a large range of concentrations.



As a sanity check, look at this website which has a description of the same experimental setup and what appears to be real data. Note that they are measuring change in absorbance of a reactant while you say you are measuring the change in absorbance of a product, $\ce{CuCl2}$