I think it's very sane!
You're doing a very good thing by feeding and trying to catch the poor injured cat.I'd be surprised if every single one of the other cats who are eating your food are strays. Even well-fed cats will often eat food if it's on offer. It's so tragic that there are so many strays though.
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When you've caught the cat you're after, if you decide to stop putting food out I'd reduce it gradually rather than stop it suddenly.