This interpretation makes no sense. The only reason to allow the player to remain when TO is called is because it doesn't cause a delay in the game - the player must be ready to go when the TO ends. The player clearly can't be ready without a TO, so no TO, you have to sub to clean the blood and cover the wound if necessary.
Making the team with the injured player call the TO suggests that it is somehow that team's fault that the player is bleeding. Frequently it is not, so the relationship between who calls it and whether you can keep the player in the game is silly IMO.
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