Launch Loop vehicle rates will be very high, perhaps more often than one 5 tonne vehicle per minute. The very low cost of launch (and the very high infrastructure costs) require high throughput handling with many automated inspection stages. It also means that missions will typically be assembled from many combined low-cost payloads in a construction orbit.
The vast majority of launches will be cargo: critical repair parts, supplies, and construction materials. Humans are launched after temporary accommodations are assembled by robots and thoroughly tested.
Repair parts will be often be unplanned, urgent and survival-critical, and must be sent fast, anywhere, anytime. The quickest method to near-Earth destinations will be redundant multistage high performance liquid fuel rockets lofted at maximum loop velocity (11.5 km/s?), perhaps further accelerated into a fast hyperbolic with an additional boost stage to perhaps 14 km/s while in the deep gravity well, then decelerated fast near arrival.