The current theory (no pun intended) is that the VBus and ground wires within the provided USB cable (which they had forgotten to bring) are thicker than standard guage and/or that the power management circuit on the input to the device has an undervoltage limit set in hardware that is set too strictly, effectively locking you out from using any standard usb cable (30AWG wires typically), even though the provided cable looks totally ordinary from the outside. I would not be surprised at this as there are many companies that make 'data block' USB cables that omit D+/D- wires and instead beef up the VBus and ground wires, but look like a totally ordinary USB cable. There are also companies that parallel the D+/D- with VBus/GND for greater current capacity, but you wouldn't know without taking it apart.
I find it astonishing that a device designed for use in potential medical emergencies could have been designed this way with no warning of this in its manual, or if if this is a hardware bug that this issue was not discovered very quickly during hardware verification tests (if indeed any were performed).
Coincidentally i was recently having issues with charging of USB power bank. Very similarly I was trying to charge the power bank with a USB A to Micro B cable from a mains USB charger and thought that the power bank was charging quite slowly. I hooked it up to a current clamp and it turns out that wiggling the micro B end of the cable when it's mated to the Micro B receptacle on the power bank swings the charge current anywhere from from 100mA to 1A. I borrowed a 'custom' USB cable that a friend had bought off Amazon and it was even worse, not budging above 300mA. Incidentally, the power bank states it can handle 2.1A charging current on its micro B input (which curiously appears to be above the specification of a micro B receptacle but that's a separate issue). Only by making a custom cable myself, using 20AWG power and ground wires, using a high quality $5 Hirose micro B plug and then epoxying the connector ends to make a tight fit was I able to get the 2.1A charging current.
I find it concerning and I'm really not convinced adding charging capability to the USB specification as early as they did was a good idea. Having said that, they were probably just trying to standardise what a lot of people were abusing USB for already.
TLDR; not all USB cables are made equal. We live in times of mass usage and abusage of USB. Be careful out there.