Crowd-sourced, randomly located vehicles in large-scale on-demand mobility systems result in uncertainty in customer waiting times, posing challenges for service reliability and system efficiency. We theoretically establish that, under nearest-eligible-vehicle assignment and the stated independence condition, any nondegenerate limit of suitably normalized customer waiting-time minima must belong to the minimum-GEV family. Under a lower-bounded spatial Poisson benchmark, we derive how the cumulative spatial intensity of eligible idle vehicles determines the minimum-GEV location, scale, and shape parameters. Numerical simulations show close finite-sample agreement between fitted minimum-GEV distributions and simulated waiting-time distributions under the specified designs, including expected eligible-set sizes as low as ten and random selection among the first three order statistics. Observations from e-hailing trajectory data provide descriptive distributional evidence that GEV distributions approximate customer waiting times. We also formulate a generalized-cost comparison across providers. A lower-bounded observed-quote benchmark links provider quotes to physical pickup-time distributions, while a behavioral benchmark uses independent minimum-Gumbel generalized-cost disturbances with a common scale and yields the Multinomial Logit (MNL) form. Moreover, we analytically derive a unique stationary throughput turning point associated with the Wild Goose Chase (WGC) phenomenon under an uncontrolled, lower-bounded benchmark. The resulting turning point provides a stress-test benchmark for examining how assignment, admission, dispatch limits, and repositioning alter stationary throughput.