Following the link given above... (And don't be afraid to open up all the categories on the left)
There are a lot of services under Custom Critical. They would certainly move a single piece. Now you'll pay for it... But paintings for example can and do use Custom Critical, they would not be a daily shipment nor bulk. But high value.
Custom Critical Air Expedite:
Any size: From Cessnas to Antonovs, we can accommodate all sizes of freight, from hand-carry items to oversized shipments.
Put it this way, something has to be important enough or these services wouldn't exist. A life/death threatening item has to be a target for this service, no? Perhaps even Temp Control services should have been used, that is ping'd back to Fedex and monitored from A to B, so if it stopped moving and/or started heat/cooling, they would know immediately and it wouldn't be stuck in some hub sort, this works outside of that. Many of these services would not be seen by standard drivers.
I get a shipper would think and they'd be right almost always that using Fedex Express would be good enough. But "almost always" perhaps isn't "good enough" in a case like this. Once you are burnt, you'll be in a conference with in-house logistics reviewing what should have happened and what is to happen moving forward. My guess is if they brought in the Fedex Sales team after a failure like this, Custom Critical would be the target.
My other guess would be if the lady were to sue, the policies Fedex has are strong and then it could be looked at as negligence with the Hospital using improper services for the critical task and the suit may change towards the Hospital.
My bottom line is UPS and Fedex (general services) are good. But they aren't critically good. Both companies have whole divisions set up for critical pieces. To me a medical product like this would fall into "critical care" and higher grade services yet should be used. I think it's at worst, learn your lesson once and then no screwing around after if you are the Hospital. When was the last time they had any problem ok'ing a cost of a product or service? The next time will be the first... Bill it!