There are a number of incidents mentioned in the post and in this thread where Red Cross workers were injured or killed by combatants. Do I have any reason to believe any of them involve "appropriation of their trademark contributed to the deaths"? No, although I would believe that the defense in most cases would be "a mistake was made". But the Red Cross (and a lot of other people) have good reason to defend those logos.
International Humanitarian Law (https://ihl-databases.icrc.org/customary-ihl/eng/docs/home):
"Rule 25. Medical personnel exclusively assigned to medical duties must be respected and protected in all circumstances. They lose their protection if they commit, outside their humanitarian function, acts harmful to the enemy."
"Rule 28. Medical units exclusively assigned to medical purposes must be respected and protected in all circumstances. They lose their protection if they are being used, outside their humanitarian function, to commit acts harmful to the enemy."
"Rule 29. Medical transports assigned exclusively to medical transportation must be respected and protected in all circumstances. They lose their protection if they are being used, outside their humanitarian function, to commit acts harmful to the enemy."
"Rule 59. The improper use of the distinctive emblems of the Geneva Conventions is prohibited."
(The Red Cross logos are such distinctive emblems.)
The purpose of these symbols is to unambiguously identify protected vehicles, protected buildings, protected people.
It's true that in some countries people slap red crosses (in particular) on stuff that shouldn't have them and doesn't need them. But that doesn't make it a good idea. In a video game in particular you could use any symbol and players would soon get the idea. How long do you think it takes Mario players to realise that one of the mushrooms makes Mario bigger, while a different one is an extra life?
Well, that settles it. Any games depicting red crosses should be patched to check if the player is shooting at vehicles or people marked with a red cross in the game, and if so, report him to the nearest police unit for an arrest.