This is all true but is missing some important detail - the box (shipment container suggests shipping container sized things, it's more of a flight case size thing) is meant for long distance transport, rather than local distribution. It holds the vaccines at -80C for ten days, and can be refilled with dry ice (which is widely available anywhere that has a carbonated drinks industry) to extend that period. The ten days is how long it's expected to be in continuous transit, worst-case. The vaccine must be thawed before use, and can be stored in a regular freezer for a few days, just not indefinitely.

Since the transport cases are reusable, and don't need to be completely populated, what will likely happen in rural areas is that they'll have a case for shipment, refill it with dry ice on site, and use a second case to transport a small number of doses to the point of use. Repeat until used up, refilling with dry ice every few days at a central location (less than a week's travel from administration sites). I don't think it's going to be a major problem in practice.