Breakover angle I think is important in that one needs enough angle to keep the "speaking length" of the string seated well over the witness points on either end.
Speaking length I'm referring to as the length of string that is drawn over the witness points at the saddle and nut and runs between them - or the long part of the string that we bring up to tension for a particular open frequency (e.g. ~41.2 Hz for an E string).
The
witness points refer to the point at which the speaking length of the string is "terminated", so the point at the nut and the point at the saddle. Strings on an instrument are not actually terminated there, but have a short length of string that breaks over the witness point, and terminates at the tuner (nut end) or bridge terminus/string thru holes/tailpiece, etc.
Functionally, the speaking length of the string is where we have the perceived tension under our fingers. This remains "the same" overall in the vast majority of instruments because the speaking length is
many units of length in comparison to the non-speaking length that goes from the witness point to the terminus on either end.
Now if we use a little thought experiment here and take this to absurdity this might shed some light. Let's say you had a 34" scale bass, but 340" long strings (yes, 28.3 feet). The string terminates at the bridge end and nut end about 153" away from the witness point. Now you have a 34" "speaking length" on your bass string, but you also have 153" of string on either side brought up to tension to support that 34" E string. I imagine (as you might too) that the 34" e string, despite being brought up to tension will be dang floppy, as there is so much string distance on either end, that the inherent flexibility of even a heavy gauge string would cause major tuning instability and the fretted note to be warbling and have other overtones caused by reverberation down those super long "non-speaking" lengths of string.
This is taken to the absurd, but can illustrate the point - for all intents and purposes when we are using short "non-speaking" length strings such that they
cannot stretch the way the absurd example above demonstrates, then the length of string beyond the witness point for the most part, becomes an irrelevant factor. The flexibility of the
speaking length has a greater magnitude of effect over the sound of the string than the
non-speaking length.
At what point the increased non-speaking length becomes
more relevant is surely something that could be measured, but it's pretty easy to see on it's face that the non-speaking length would likely have to be pretty darn long (many, many inches) in order to start having an influence on the speaking length of the string in the system as the thought experiment explains above.
So what does this all mean? I think that yes, how the string is terminated beyond the witness points is important to the system, but what is the magnitude of the overall influence it imparts? Seems to be it would be negligibly small, and that the most important factors here are clean breakover angles at the witness points so that the string does not break or wear unnecessarily, and that beyond that, it's aesthetics and simple functionality. The Yamaha example possible increases string life by having a less steep angle at the witness point, but the hundreds of examples of 30 year old strings on string thru basses probably also shoots that argument down.
Regarding the double bolts at an angle in the neck pocket - it certainly looks cool and helps keep the neck seated well, but I would also suspect that a well-matched neck/pocket pair is also just as effective, as also evidenced by many thousands of perfectly functional basses without this innovation.
Jeez, I've been working far too many days in a row at this point...
