Weird, I am very aware of the ethical concerns with kosher and halal slaughter but never heard about health concerns. BTW, the Amish way to slaughter a chicken is a bit more alarming than the kosher method.
Actually poultry in particular is where the health concerns exist with Kosher and Halal slaughter
...due to the rules that heat and steam can't be used for feather removal (meaning they have to use more aggressive "plucking methods" which open up the skin and make it more prone to infection)
And there's mixed research about the topic of the spiked cortisol levels in the ruminant animals at the time of slaughter for animals that have to get their throats cut while they're still conscience vs. one that get stunned ahead of time, and the potential impacts on the quality and shelf-life of the meat itself after the fact.
Why the cultish obsession with "the animal has to feel the pain" even exists, I have no clue...seems like some evil stuff to me. And I suspect, that in this current time, if that was only a feature of Kosher slaughter and not Halal, many more would vociferously come out against it.
And it goes without saying that wet market food culture brings up all kinds of health risks (and ones that don't always stay contained to regions where it's taking place). Common sense would dictate that butchering snake meat on a table with the remains of who knows what, that's been sitting out in the hot sun next to a bunch of diseased animals, is a terrible idea.
I've literally been called xenophobic for suggesting that there should be an international effort to shut down wet markets.
"Just because it seems weird to you, who are you to say that the Chinese and Sudanese people have to change their cultural traditions", to which my response has typically been "when they start creating respiratory viruses that don't stay contained to the region, then it becomes my business"