Not a bad idea if you have a flatbed trailer laying around unused

Trailers are useful and thus expensive.

General assumption maybe I shouldn’t be making, is that the trailers are already owned by the organization responsible for this and would already only be used for this purpose.

Perhaps the fact that you could build these theoretically faster than doing on a trailer makes it a moot point, then again if trailers are your bottleneck maybe it’s possible to schedule building around that to keep delivery smooth

Trailers generally require vehicle registration to be towed on the road, and tow vehicles are relatively expensive. Despite already being on wheels, individual residents would still be unable to move their shelter themselves in response to being told to move on. And any organization that helps out with tow vehicles creates a centralized target to attack.

The solution in the OP of building this as part of a dedicated tricycle is pretty damn slick. "Yes of course officer I am moving along, as fast as I can"