It's a great post. Recently i'm interesting in the stringing machine too. And read some docs about that.
For the dropweight machine.
If considered the string tension lose eg. the string will be elongated. For example sofer strings. The dropweight will not be a accurated system. In my pic, left, the string is pulled at a constant tension, and continuously whether the string is elongated. Right, if the string is elongated eg. tension lost, the lever will not still holding the horizontal position. For example, extreme situation if the string losed too much, when the Q=-90degree, the string tension is 0. Stringer have to adjust the tension for the tension lose. Increase the Q between the arm and the string gripper. One hand hold the 1-2kg lever, and one hand hold the string gripper, and the eye have to watch the indictor. It's complicate operation. I saw some video, if the stringer need to adjust the Q, at least adjust 2-3times.
To avoid the string elongated, many stringer using Pre-stretch method. It's easy to control.
I agree with your opinion about the F=ma, the speed should not be fast. and your example catapult, I always using a manual Manual crane for example, it is easy to find in a car repair shop. And easy to use to elevate a car. I think that is why someone adjust the Q many times then the tension is much higher than correct.
If the string is hard to be elongated, the tension is excatly accuracate, if the string is softer, the tension maybe need more experience to adjust.
The Stringway machine invients a method to make "The tension is the same for every angle of the lever" It's great, but I don't have that machine, wish to learn how they do that.
Learned much from this topic and reply, it's great.