So you're just going to ignore the later part where Josuke says "do I look like a doctor to you?" If Kira was actually a normal dude, he'd just ask Josuke to call an ambulance, not leave his life in the hands of two tough-looking teenagers. Like, what could a 16-year-old possibly have on his person at all times to help? Band-aids? Gauze? Not likely. Possible, sure, but not nearly likely enough for Josuke to consider. Was it a 100% logically-sound plan? No. But Josuke isn't a 100% logically-sound character, anyways, so it doesn't matter. Is Josuke a clever kid who would take a normal person seemingly knowing about Crazy-D as proof enough in a tense moment to assume he's the killer? Yes, and that matters.

Actually, Josuke was playing one on Kira. He said that just to know how Kira would react. Josuke is a high schooler, he can't "patch people up",. Understand now?