The biggest mistake that can ever be made by any civilization is to tie anything that can save or better the life of people to one institution, doesn't matter which one it may be.
In general notion trusting religion for anything is stupid anyway, even (or rather especially) when magic and gods are proved to be real.
@Despada
I see it differently, the problem I see is that no back up has been developed because it is considered "heretic" to cure diseases if it is not by using the "power of God" which then stop any development on medicine, which could be used much more efficient as it would be something anyone could use instead of having to rely on some specific person from the "official organization". One example of that problem is the world of overlord where even if someone has the capacity to heal someone else they are not allowed because healing must be done through the temples.