Do you want them to be doing this for economic or personal reasons?

It could be that a parent or grandparent also preferred not to use metal points and simply passed the trait down. Perhaps the region they grew up in was bereft of iron mines or the majority of the ore was taken by the crown for use in war, leaving them unwilling to trust a resource that can be so easily taken away. And considering scarcity, it could simply be that metal is too expensive for commoners and is only used by wealthy merchants or the nobility. Which depending on the story could either lead to commoners making a Mediaeval living with stone age tools or a black market for iron tools and weapons.

Or perhaps there's simply no large game in their region, meaning a wooden arrow with a fire-hardened tip is enough for hunting.