How do ranged weapons invalidate persistence hunting?

Even with a modern bow it's still really difficult to sneak close enough to a deer to reliably make a kill shot. You're not going to sneak close enough to poke it with a spear and with game that size, throwing rocks is not really an option either because that wont kill it. Something like axis deer is quick enough to even dodge a modern arrow.

The reality is that the animal will notice you and it will out-sprint you as well but it wont outrun a human on a long distance. When the animal is exhausted and no more able to run, then you can then stick your spear in it.