Light is a powerful non-invasive tool that can be exploited to probe highly scattering media like biological tissues for different purposes, from the detection of brain activity to the characterization of cancer lesions. In the last decade, time-domain diffuse optics (TDDO) systems demonstrated improved sensitivity when using time-gated acquisition chains and short source-detector separations (SDS), both theoretically and experimentally. However, the sensitivity to localized absorption changes buried inside a diffusive medium strongly depends on many parameters such as SDS, laser power, delay and width of the gating window, absorption and scattering of the medium, instrument response function shape, etc. In particular, relevant effects due to slow tails in the instrument response function (often due slow decaying phenomena occurring within the detector) were noticed, with detrimental effects on performances. Here we present a simulation study based on the Radiative Transfer Equation under diffusion approximation, using perturbation theory under Born approximation. To quantify the system sensitivity to deep (few cm) and localized absorption perturbations, we exploited contrast and contrast-to-noise ratio (CNR), which are suitable standardized figures of merit internationally agreed among different research institutions. The purpose of this study is to determine which parameters have the greatest impact on these figures of merit, thus also providing a range of best operative conditions. The study is composed of two main stages: the former is a comparison between simulations and measurements on tissue-mimicking phantoms to validate the model employed, while the latter is a broad simulation study in which all relevant parameters are tuned to determine optimal conditions for each one. This study essentially demonstrates that under the influence of the slow tails in the instrument response function, the use of a small SDS no longer corresponds to optimal contrast and CNR. This work sets the ground for future studies with next-generation of TDDO components, presently under development, providing useful hints on relevant features to which one should take care when designing TDDO components.