I personally think that areas of unmowed grass are way overused on modern courses. (Jeff's second picture made me shudder.)
Sometimes that's what you're starting with and it makes sense to leave it. Barnbougle was 100% marram grass to start with, and we only cleared as much of it as necessary to make the course playable, nearly all of which is mowed at fairway height.
Some of it has to do with technology and safety issues. The farther everyone hits the ball, the more land the course requires, and the more pressures from the cost of maintenance to leave some of the ground "unmaintained". It probably would have been better if this had just been left "undisturbed" as well, but as new courses move further away from good natural sites, nobody wants soybean fields between the holes and long fescues seem to be the only solution anybody can think of. (Not so imaginative for a bunch of trained landscape architects, is it?)
There are also awkward spaces between one fairway and the next to allow a safety buffer for golfers and for the cart paths ... you rarely see a modern course where you can hit it into another fairway, as you could on nearly all courses before 1930. Personally, I think the bigger spaces between the holes are overkill, and that's a big part of the "cause" of this trend.
To me, the solution would just be to put some minimal maintenance on these areas. Keeping them lean and mean (not fertilizing or watering much) and gang-mowing them once every 3-4 weeks in season (until they start shutting down because of drought stress) would avoid the unplayability issues and really wouldn't cost a heck of a lot. But members will start complaining that it looks weedy ... that's the other part of the reason you see ridiculous amounts of hay, because if it's turf instead the members can't help themselves from wanting it to look perfect.
Back when I worked for Pete Dye, he was experimenting with Centipede for southern roughs. It provided a great textural contrast, and it grew really slowly so it cost very little to maintain. But it disappeared at Long Cove because irrigation and fertigation overspray, and golf cart traffic, wiped it out.