Niamh is an archer and I always look through her eyes when we are in combat and I can honestly say she has never fallen off of a cliff, or indeed anything as a result of not being able to see her feet.
I would suggest it's an immersion breaker for you - because it's not an immersion for everyone.
It depends how you use first person. I agree that using a bow in third person if far more difficult, especially when it come to needing to be accurate with distant targets.
I play first person all the time because I try to "become" the character rather than be just an onlooker, (full immersion). I find that things don't get so impersonal that way, I can use my own moral compass to act in a way that I would if it really were me there. I don't just kill someone who annoys me or Belethor's General Goods in Whiterun would be top of my list, his condescending "Do come back." as if I was being a total skinflint annoyed me no end. I use his shop now only when I need to sell off the junk from quests. Yeah, I know it's only a game but I do like to roleplay so I take it personally as I would in real life. When you immerse yourself to that sort of level, certain things tend to break the immersion, not seeing your own feet when you look down is one of them.
I have played first person games where you can look at yourself in a mirror, look down and see your own legs and feet and even one where your hair can cross your peripheral view in a windy area or when you look down far enough. That level of detail allows you to get into the game and really become the character. I even get an adrenalin rush when it comes to melee combat and trying to escape capture by guards etc.