Although largely an academic reply here I'll add a variable as well.
Player attack.
Some of the guitars that we set-up are on the cusp of needing neck resets. As such the saddle has already been taken all the way down, string ramps may have been milled into the bridge depending on what's appropriate for the instrument... and these days bridges are rarely shaved as this practice can be, at times... frowned upon. Again for bridge shaving it's what's appropriate for the instrument. An important term in our business.
Anyway David's 5 degree minimum is a great number and I would consider that as a minimum or near minimum. A player with a light attack will not drive the string as much as a player with a moderate or heavy attack. Or, at times, a player with a light attack who drinks too much....
But without digressing too much the vibrating wave confined by the boundaries of the "speaking length" of the string may breach the saddle and/or nut slot if the break angle(s) is not substantial enough. Hence the question here although seemingly an astetic question is really a physics question.After all head stock set-back angle is important to the physics of vibrating waves or back to that 5 degrees that David cited.
Some years back I privately approached a builder here who was building electrics with only 2 - 3 degrees of break angle over the nut. My concern was that this would not be enough and his design for his head stock would need to be augmented with string trees. He didn't agree and even though I was trying to be helpful I think that I pissed him off. It was also a private, PM exchange because I didn't want to embarrass anyone.... This never made it to the general forum. Oh well, the best laid plans..... and so on and so forth...
Anyway Frank Ford is fond of splitting the difference of say a 14 - 15 degree headstock for the nut slots or 7 to 7.5 of break angle. This works great by the way and is my go-to standard when possible. But it's not always possible especially in the Fender bolt-on neck world where a Strat's G string may have only 3 degrees of break angle due to Leo not understanding guitar physics regardless of his mastery of manufacturability. An additional string tree or many more winds on the tuner can greatly improve this. By the way there is a tip for you guys. You can increase break over angle at the nut by employing more string winds on the tuners. I sometimes use very nearly the entire, uncut string on Strats for the G string to improve things.
Lastly the reason why your head stock set-back angle matters although it does contribute to our perception of what a guitar should look like is far more important than just another pretty face.... plate.... We are setting the stage for the physics of break angles for strings to be possible.
Break angles matter because after all guitars are tools for musicians.
