The interface between a metal and a semiconductor that have dissimilar phonon frequency spectra typically exhibits a low thermal boundary conductance, as found for many Au/semiconductor interfaces. A thin metal layer (i.e., a contact or adhesion layer) placed between the gold (i.e., the capping layer) and the semiconductor can increase the thermal boundary conductance, an effect that has been attributed to the contact serving as a vibrational bridge. The impact of a change in the electron-phonon coupling due to the presence of the contact on the thermal boundary conductance, however, is not fully understood. To assess the roles played by vibrational bridging and electron-phonon coupling, we apply a two-temperature non-equilibrium molecular dynamics simulation approach. By specifying the heat flow through a structure, we can predict the temperature profiles in the electronic and phononic subsystems and the resulting thermal boundary conductance(s). Four types of structure built from metals (M) and semiconductors (SC) are considered: M/SC, M/M/SC, M/SC/SC, and SC/SC/SC. Increasing the contact layer thickness for the M/M/SC systems results in a monotonic increase in thermal boundary conductance, matching trends from previous experimental studies. The phonon density of states in the contact layer identifies the existence of non-bulk phonon modes for small thicknesses. Increasing thickness results in contact layer phonons gradually shifting to bulk frequencies, which explains the occurrence of the plateau in thermal boundary conductance for large thicknesses. A comparison between the M/M/SC, M/SC/SC, and SC/SC/SC systems provides insight to how the electron-phonon coupling in each metal layer impacts thermal boundary conductance.