It could not matter less that the advice given in these books used to make sense at some point.
This assumes literally everything is different today. Which of course is nonsense.
How could you use "hygiene" to justify the unnecessary torture of animals with outdated butchery techniques? 2000 years ago there probably were great health reasons for doing so. We're in 2021, there's simply no excuse.
Some religions evolve faster than others. I would hope the different faiths can come up with iterative improvements to their standard operating procedures that still meets the requirements. I believe the Jewish religion even has a concept about creatively outsmarting the religious rules as being a virtuous thing (eg women wearing wigs).
But my question still stands, specially on the light of such view of religion: if religion is, these days, meant to give people internal piece, does it need to dictate such practical things as how you should kill animals?! How do you reconcile that with your view on religion?