Pikes were used much the same way right? Surprised I never put the two together, ancient humans weren't stupid so of course they'd realize that was a better way of causing harm than just throwing it. Not to mention their use of leverage in weapons like the Atlatl. No clue on the timespan of these things but I do find this stuff interesting.
Honest answer, usually animal sinew, or certain grasses could be used as well. The nice thing with string, once it was figured out was you could make as much as you could, and make it as long as you wanted.
I totally count sinew as string. That probably led to plant string. Think about really fine string, or thread, and think about how many miles of it you carry around on you every day. It's crazy how taken for granted it is!