I doubt you'd have enough control or the right vantage point to strike a killing blow from the end of a rope. You need to puncture vitals or you'll just end up with a very angry mammoth with a spear sticking out of it. So "steering" the spear to the point of impact and reacting precisely in the last fraction of a second is important. A modern corollary might be trying to shoot clays by steering your shotgun with a rope.

So long as you can survive the encounter, if you inflict any kind of decent wound, wouldn’t the animal eventually die due to blood loss or infection? Not at metal as an immediate kill, but whatever works.

Even hitting vitals won't be "immediate", you'll still have some hectic moments/minutes dodging the thrashing mammoth. But if you don't strike a mortal blow the danger is much greater because the animal won't weaken and slow down. Instead it'll get pissed and probably fight back even harder.

I think rope, at least of any appreciable length and strength, is a much more modern technology.