I think only the length of the scale determines tension for any particular gauge/weight of string. So the distance between the nut and saddle a determinant, but the distance from saddle to stop bar is not part of the equation.
There are a certain amount of harmonics present on the string outside of the scale length, and this can have an effect on sound. The string between tuner and nut, and between the saddle and tail piece are audible if plucked, and a certain amount of sympathetic vibration happens when the intentional note excites the string beyond the nut and saddles.
I've occasionally stuffed the head stock of basses and a guitar or two to stop the weird ringing overtones that come that. It seems more apparent to me on bass, FWIW.
I'd tend to think that the finger tailpiece was probably designed for appearance primarily. It sure looks nicer than a trapeze tail.