There's plenty of studies that TNR works, and even more that show that kill methods don't.  Keep looking.

It's a false dichotomy because monk seals aren't dying from cat poop.  A few monk seals and a couple birds does not a pandemic make, even if someone actually was able to prove direct causation, which no one has.

You can say goodbye to our endangered species anyway because of global climate change.  The malaria epidemic is growing while you try and declare a war on cats