I think the worry is a broken stick has a chance of producing something sharp that can injure others. I think it is more of a worry when the break is along the shaft of a wooden stick (as you could stab someone, you're basically wielding a spear) but even a split along the bottom of a wooden stick's blade has the potential to lose splinters.
I played in Canada, but this ruling seems quite similar to what was in effect for us: https://www.usahockeyrulebook.com/page/show/1084649-rule-605...
I think the worry is a broken stick has a chance of producing something sharp that can injure others. I think it is more of a worry when the break is along the shaft of a wooden stick (as you could stab someone, you're basically wielding a spear) but even a split along the bottom of a wooden stick's blade has the potential to lose splinters.