It sounds to me like building logistics pipelines between hospitals (even partnering with the healthcare pneumatic logistics companies to do this) would be an ideal way to get started. Then expand the network from there with QoS packet prioritization schemes. Similar to how large Fiber Optic networks are done in some regions, or how arpanet started.
I used to work in a hospital lab, it's a lot of biohazard stuff being moved around so probably best it stays a closed network. The pipedream idea is great. However I don't get why they focus on food/grocery delivery. I'd want them to replace a majority of delivery trucks of the Fedex/UPS/Amazon variety if I was going to allow them to tunnel a citywide network. Quantifying the reduced wear and tear on city roads may even make it an easier sell to local governments.
It sounds to me like building logistics pipelines between hospitals (even partnering with the healthcare pneumatic logistics companies to do this) would be an ideal way to get started. Then expand the network from there with QoS packet prioritization schemes. Similar to how large Fiber Optic networks are done in some regions, or how arpanet started.