We present a comprehensive description of electrically-driven vertical-external-cavity surface-emitting diode lasers (VECSELs) at 980 nm, mode-locked by saturable absorber mirrors. A novel partially-integrated time-domain model combines accuracy and flexibility, allowing for a semi-analytical stability analysis of the compound-cavity modes, tracking the mode-locking onset and an optimization analysis. The linear stability analysis of the monochromatic solutions (i. e., the compound cavity modes) indicates that single mode solutions exist and are stable only in a limited current range around threshold. Increasing the current above this current level leads to a multimode solution through a Hopf bifurcation. This bifurcation point is followed by a continuous transition leading from harmonic oscillations to fully-developed pulses that correspond to the mode-locked solution. We obtain stable, fully-developed mode-locked pulses of few tens of picoseconds at 15 GHz repetition rate in good agreement with reported experimental results. We discuss the dependence of the mode-locking regimes on the reflectivity of the distributed Bragg reflectors, spot area of the spatial mode, and number of quantum wells in the emitter and absorber cavities. The optimization analysis reveals that, in order to favor the mode-locking onset, the effective coupling between the emitter and saturable absorber cavities has to be optimized through the standing wave pattern in the composite cavity and spot-area of the spatial modes.