With the sheer amount of leverage that can be gained from using a spear-thrower and its extremely cheap cost, why did people stop using them in warfare, opting to haul large spears with their bare hands instead?
The successor of the spearthrower appears to be the bow. The bow has four advantages:
Longer effective range.
Arrows are typically smaller and lighter, and if a quiver is used, more of them can be carried.
The arrow is faster, and more difficult to dodge or parry.
The bow is more effective at very short range. The motion of using a spearthrower give the target a very good idea as to where the spear is aimed, and when it will be released. This gives additional warning that might be used to dodge the attack. In addition, the bow can be used effectively while standing still, and the spearthrower should make use of body motion forwards. This means that the thrower moves towards the target while throwing, which might be undesirable when that target is very close and has a thrusting spear in hand.
It might also be the case that the bow is easier to use on horseback. Not having seen a spearthrower in use from horseback, this is speculative, but the importance of using the bodyweight and forward motion for maximum power when using a spearthrower (which will be impossible to do fully on horseback) suggests that this would be the case.
The other competitor is the hand-thrown spear. A hand-thrown spear allows better use of a shield while throwing, so use at very close range is safer. Further making it safer at very close range, one can simply use the spear as a thrusting weapon if the enemy is too close. But the extra range due to a spearthrower makes it unlikely that the hand-thrown spear will replace the spearthrower in warfare if the spearthrower is in common use in that society. The hand-thrown spear can be a useful supplement to the spearthrower, sling, and bow, and was used alongside all three weapons.
On a technical note, "leverage" is a poor explanation for the functioning of a spearthrower. The fundamental principle of the lever is "work in = work out". That is, the force applied by the user, F1, times the distance the user moves the end of the lever that force is applied to, d1, is related to the force exerted by the other end of the lever F2 and the distance it moves through d2 by:
F1d1 = F2d2.
As a result, a lever allow you to apply a larger force (if d2<d1), but the work done (F*d) remains the same (the earliest scientific explanations of the lever used this idea, in the form of the principle of virtual work - the discussion in the pseudo-Aristotlean Mechanica is very nice). If "leverage" was the full explanation of the spearthrower, it wouldn't allow the throwing of a spear with greater speed. The key to the spearthrower is that the force that a thrower can exert on an object during a throw is limited by how fast the arm can move (both by the speed of the muscles and the work used to accelerate the arm). The spearthrower allows the arm to move more slowly relative to the spear, compared with a hand-thrown spear.
This means that the spearthrower gives the most benefit with light spears. With very heavy spears (e.g., if you want very heavy javelins to maximise armour penetration), there is less advantage. Since such spears are often thrown just prior to entry into hand-to-hand fighting (because of their short range), hand-thrown can be better than a spearthrower.