The origin and progenitors of short-hard gamma-ray bursts (GRBs) remain a puzzle and a highly debated topic. Recent Swift observations suggest that these GRBs may be related to catastrophic explosions in degenerate compact stars, denoted as 'Type I' GRBs. The most popular models include the merger of two compact stellar objects (NS-NS or NS-BH). We utilize a Monte Carlo approach to determine whether a merger progenitor model can self-consistently account for all the observations of short-hard GRBs, including a sample with redshift measurements in the Swift era (z-known sample) and the CGRO/BATSE sample. We apply various merger time delay distributions invoked in compact star merger models to derive the redshift distributions of these Type I GRBs, and then constrain the unknown luminosity function of Type I GRBs using the observed luminosity-redshift (L-z) distributions of the z-known sample. The best luminosity function model, together with the adopted merger delay model, is then applied to confront the peak flux distribution (log N-log P distribution) of the BATSE and Swift samples. We find that for all the merger models invoking a range of merger delay timescales (including those invoking a large fraction of 'prompt mergers'), it is difficult to reconcile the models with all the data. The data are instead statistically consistent with the following two possible scenarios. First, that short/hard GRBs are a superposition of compact-star-merger-origin (Type I) GRBs and a population of GRBs that track the star formation history, which are probably related to the deaths of massive stars (Type II GRBs). Second, the entire short/hard GRB population is consistent with a typical delay of 2 Gyr with respect to the star formation history with modest scatter. This may point toward a different Type I progenitor than the traditional compact star merger models.