Sum of a rate constant gives the total decay rate
I am simply confusing two different variables here, rate and rate constants. 

In this situation, the sum of the rate constants is the sum of the individual rate constants of the processes occuring in the reaction i.e. 

$$\ce{k_F + k_{ISC} + k_{IC} + K_{PC}}$$

This is consistent with my current knowledge. 

It should be referred to as "total decay rate constant"

Of course, the rate equation can still be written here as:

$$\ce{v_{rate} = k[A]}$$