That sounds really annoying. Imagine leaving a town fully stocked, get into a fight, roll a nat 1 on first attack and immediately have no arrows for the rest of the fight. What, did the ranger just forget that the quiver was empty before leaving town?
I've always disliked the concept of critical fails in general, and this is a great example of why. If we're to believe that our characters are truly these great warriors with far more skill and experience than an average person like the texts usually say, how does it make sense for these professionals to just completely blunder 1 in 20 of their attempts at everything? From an RP standpoint, it doesn't add up, and from a gameplay standpoint, it's just annoying as hell IMO.
This reminds me when I was trying to fish out natural ones as a new DM from my players. They were rightfully annoyed by this. I fortunately grew from this and no longer fish natural ones
People tend to not realize how often something is going to be happening with a one in twenty chance and that you are going to be rolling your basic attack roll a gazillion times per session. When you start rolling the dice, making attacks every turn, that is going to come up very often. In fact, statistically this rule would mean that your character would be carrying on average ~13½ arrows. By the time you've rolled 14 times it's more likely that there was at least one 1 in there than not. With multiple attacks per turn that's going to happen infuriatingly often.