I remember somewhere PJ explaining why he made that change. It was something to the effect that while Strider carrying Narsil is dramatic and romantic, it’s not at all practical for someone who wanders the wilds. A broken sword is not nearly as useful in a fight as a full-length sword. Plus, it makes sense that such an important sword would be kept in a place of honor rather than being carried around in the wilds where it could potentially get lost or damaged further. So, while it’s not what Tolkien wrote, it does have a logic to it.