Does a child's toy, which is supposed to represent a first aid kit, with the red cross on it constitute a misuse?
Can a random private hospital not use the red cross?
The important thing about the Red Cross and its brand is that they are neutral. The Genova convention declared they get a special marker, and a rule against harming them in wartime. A US army doctor presumably does not seek out to treat both sides of the conflict, and does not get the special protection the rules of engagement afford to the Red Cross. Nor would some random soldier carrying a J&J first aid kit get any protection. In particular the risk is that the more common that symbol is, the less distinctive it is, a particularly troublesome effect during armed conflict where decisions about where to point a rifle and whether to pull a trigger are being made rapidly.
> Can a random private hospital not use the red cross?
A random hospital definitely cannot, without permission (and presumably, some covenants). And it'd definitely not be an enforceable trademark on their end so not a smart branding move anyways. It's usually not a huge deal -- in the US the hospital sign is blue with a big H. In video games you can just use red background with a white plus (but thats like, the swiss flag) Or in the case of TF2, a red (or blue) cross on a yellow circle.
It would likely help their cause if there were an alternative public domain recognized symbol. The ISO standard is apparently White cross on green background: https://www.iso.org/obp/ui#iso:grs:7010:E003 but pretty much nobody knows that.
FWIW, first aid kits in the UK almost exclusively use that symbol.