Calculation of line of sight system gain I'm trying to calculate the overall gain of the transmitter-receiver system for a line-of-sight wireless transmission.

My question is how to calculate the overall gain of the system. Is it as simple as calculating the gain of the transmitter and receiver separately and then adding them together?
You need to multiply the antenna gains, not add them. Specifically, if the free-space loss (attenuation) is $L_{FS}$, the transmitter antenna has gain $G_T$, and the receiver antenna has gain $G_R$, then the total system loss $L$ is $$ L = \frac{L_{FS}}{G_T G_R}. $$ The system gain $G$ is $$G = \frac{1}{L} = G_{FS}G _T G_R, $$ where $G_{FS}$ is the free-space gain.

Of course, if you're doing the calculation in decibels, then the antenna gains are added: $$ G_{dB} = G_{FS,dB} + G_{T,dB} + G_{R,dB}. $$