This paper provides a conceptually simple, memoryless proof to the capacity of the Anantharam and Verdu's exponential server timing channel (ESTC).  The approach is dual to Rubin's approach for characterizing the rate-distortion of a Poisson process with structured distortion measures, and illustrates that the ESTC is ``essentially'' a memoryless, additive noise channel.    This approach obviates the need for using the information density to prove achievability, by exploiting: 1) the numerical entropy rate of any finite-rate point process tends to $0$;  2) the ESTC channel law is memoryless when conditioned upon a process whose entropy rate tends to $0$.  Given these observations, achievability is shown using a standard random coding argument, the second law of thermodynamics for Markov chains, and the multiterminal AEP.  A converse is also provided that exploits how the ESTC has an asymptotic additive-noise structure, and uses the maximum-entropy nature of the Poisson process - thus mimicking the AWGN converse.  Lastly, we conclude with a statement about a general class of point process channels for which we can characterize the capacity, and show that our converse methodology provides a non-trivial converse for the tandem queue.