@BaileysOffroad Horizon Zero Dawn solved this years ago. That game has quiver limits for all of the bow weapons, and it works well.
Also, from a physics perspective, arrows most certainly should hurt less than a heavy or sniper round. High-velocity munitions carry much more energy than an arrow (arrows glide, bullets don't). The impact damage from a single AR round can completely scramble a person's torso. Arrow damage is relatively localized, whereas bullets can tunnel, tumble or expand on impact. Unintuitively: with a high-velocity bullet, it's not the piercing that [usually] kills, it's the energy behind the hit.
