While what you are saying does make a certain amount of sense, there is one thing that occurred to me. Why would NASA use these objects if not the increase speed? You actually gave the answer, but didn’t elaborate on it. I think it’s interesting, so I will.
When we send a vehicle out into deep space, we are trying to send it to a target object. That target object may be traveling horizontal to us. It may actually be coming in our direction. Using the sling-shot effect, can help us achieve the best angle for the approach. It’s an angle that we couldn’t obtain on a straight shot path from Earth to the object without expending far more energy than we have on board.
This is a way to achieve that without increasing the delivery systems (and I’m not even sure that could achieve what we need to), and allows us to cut cost as well.
Actually, at the time of the launch the object is probably going away from us. So a straight line trajectory from earth includes the angular velocity of earth at the time of launch which unless the ship was launched at the perfect mathematical time, will not direct it in a vector that will intercept the target without burning a lot of fuel. But if you can use earths angular momentum and direct your craft to another object that will just happen to intercept your ship, then using small thrust manuevers you can redirect your ship to take advantage of a gravity well, to redirect your vector years after takeoff in a direction towards your goal with almost no fuel. The fuel use is trying to overcome earths angular velocity, which you get for free banking around a planet.