I think if the welfare of the prey is taken into account and there's no real difference there, then I would have nothing against any competent person bow hunting. But personally, it's not something that I would do.
It's all very well to say that successful bow hunting has been done by many before the advent of firearms, but it is significant that when firearms became readily available, bow hunting became a minority thing. There's a reason for that.
Here's something to consider though: How about hunting deer with a .22LR? I would guess that pretty much everyone on here would hold their hands up in horror. But, like a bow, if you stalk close enough and are accurate enough, you would achieve a clean kill. (before you think otherwise, I don't advocate .22LR for deer). So where's the difference?