The discovery of GW170817 and GRB 170817A in tandem with AT 2017gfo cemented the connection between neutron star mergers, short gamma-ray bursts (GRBs), and kilonovae. To investigate short GRB observations in the context of diverse kilonova behavior, we present a comprehensive optical and near-IR catalog of 85 bursts discovered over 2005-2020 on timescales of <~12days. The sample includes previously unpublished observations of 23 bursts and encompasses both detections and deep upper limits. We identify 11.8% and 15.3% of short GRBs in our catalog with upper limits that probe luminosities lower than those of AT2017gfo and a fiducial neutron star-black hole kilonova model (for pole-on orientations), respectively. We quantify the ejecta masses allowed by the deepest limits in our catalog, constraining blue and "extremely blue" kilonova components of 14.1% of bursts to M_ej_<~0.01-0.1M_{sun}_. The sample of short GRBs is not particularly constraining for red kilonova components. Motivated by the large catalog, as well as model predictions of diverse kilonova behavior, we investigate modified search strategies for future follow-up to short GRBs. We find that ground-based optical and near-IR observations on timescales of >~2days can play a significant role in constraining more diverse outcomes. We expect future short GRB follow-up efforts, such as from the James Webb Space Telescope, to expand the reach of kilonova detectability to redshifts of z~1.