Melvyn,
Great topic. You are absolutely correct in PP2 about the "handy" vs. "hidden" debate, and in pp3 that they have a major impact on design. I once started a gca class I was giving at the Turf show what was the first thing I designed, and then shocked the participants by saying it was the cart path. While that is not absolutely true, the fact is that its such a major part of the circulation of a golf hole that it must be considered if not first, then concurrently with the design of the hole.
Simplistically speaking, we need a few hundred feet of open area in a few spots along the fw and near the green for path to hole access. It isn't wise on a high play/high cart useage course to block those access routes with bunkers or mounds. doing so introduces a lot of "cow paths" with wear and tear and compaction. So, yes, there is an impact.
That said, I feel that it is just one more thing to work into the design, with environmental issues, etc. Since I have never had the opportunity to design a course with no paths (installed or planned for the near future) I just don't look at it as you do. Maybe that is sad, but for me, its a fact of life and that horse is out of the barn. As to visual distraction, I think if the path fits the landscape well it can look like a country road in a pastoral landscape. That isn't nearly as a big a negative to me as power lines, houses, warehouses, etc. that also clutter up the view on a typical gof course.
In some ways, I actually do enjoy designing the path. Taken as its own design problem of providing ease of circulation, using them as drainage cut offs, hiding or making attractive with flowing curves that match the landscape, etc., it is a fascinating sub problem in design. While there are many ways to design a good strategic hole, its pretty much a given that human nature inevitably makes people drive the shortest line to where they are going.
Has it ever stopped me from placing a bunker exactly where I wanted it? Sure. Have paths ever been the straw that broke the camel's back? I don't think so. Just because bunker can't be placed at "X" on a particular hole doesn't mean that there are no other ways to design a good hole with bunkers elsewhere, nor does it impact every hole. If a designer was so inflexible as to not be able to consider multiple options of design they wouldn't be much of a designer anyway. Good design is really the result of a trial and error process of floating a few ideas, refining the best ones all things considered, and then refining some more. You just can't throw up your hands when any condition - nature, cart paths, enviro regs, owner's needs, etc., prevent you from designing one particular pet idea.
As John hints, in 40 years of golf, I can count only a few times where I have seen a ball hit a path and affect play, albeit, one of them was the last time a played. So, it does happen, but when you consider 30,000 uses of a cart path in a given year vs perhaps 30 or even 300 (about 1 a day) negative instances of a ball deflecting badly off them, the cost value ratio to a golf course owner is pretty high.
Anyway, that is my $0.02. Your posts advocating going back to the strong traditions of golf have had me wondering lately about the whole concept of just how much golf should change with the times. I guess it has no more immunity from change as the rest of the world's activities and situations, does it? On the other hand, I can see the appeal of playing it old school. While you might not like it, I see the most likely outcome as some golf operator sponsoring "throwback day" for special occaisions (complete with walking, no GPS, rangers in knickers, etc. ) in effort to create some marketing buzz, vs a wholesale return to walking.