Pseudomonas aeruginosa move across surfaces by using multiple Type IV Pili (TFP), motorized appendages capable of force generation via linear extension/retraction cycles, to generate surface motions collectively known as twitching motility. Pseudomonas cells arrive at a surface with low levels of piliation and TFP activity, which both progressively increase as the cells sense the presence of a surface. At present, it is not clear how twitching motility emerges from these initial minimal conditions. Here, we build a simple model for TFP-driven surface motility without complications from viscous and solid friction on surfaces. We discover the unanticipated structural requirement that TFP motors need to have a minimal amount of effective angular rigidity in order for cells to perform the various classes of experimentally-observed motions. Moreover, a surprisingly small number of TFP are needed to recapitulate movement signatures associated with twitching: Two TFP can already produce movements reminiscent of recently observed slingshot type motion. Interestingly, jerky slingshot motions characteristic of twitching motility comprise the transition region between different types of observed crawling behavior in the dynamical phase diagram, such as self-trapped localized motion, 2-D diffusive exploration, and super-diffusive persistent motion.