Regarding trail use impact and the R2R2R …
I hiked/ran the R2R2R in 2011. Late April, perfect weather (20 overnight, 40s on the rim, 70s in the ditch), and about 50 of us R2R2Rers (I was there on my own after an aborted PCT hike due to—get this—too much snow, everyone else happened to pick the same day). The day before I went to inquire about the trip at the backcountry office given the number of signs which said "DO NOT ATTEMPT TO HIKE TO THE RIVER AND BACK IN A DAY." The issue, of course, is that unprepared hikers with no water and a few extra pounds would have no issue going downhill 7 miles, but would then be overwhelmed by going uphill in hot temperatures and require evacuation. (This is not the case with most hiking trips, where the climb weeds out the unprepared at the start and the weather gets cooler towards the top.) Their response: "if you've done enough research to be asking these questions, you'll be fine."
A group of us left around sunrise and were at the river by 7 a.m. and were the only ones on the trail going down. Beyond Phantom Ranch, there were just the R2R2Rers: few backpackers, no day hikers from the North Rim. Most of us ran alone or in small, impromptu groups, everyone carried enough water, and there was great camaraderie: chatting about ultras while jogging along Bright Angel Creek. Then it was back to Phantom Ranch, and a slog up the Bright Angel Trail (late enough in the day that I didn't run in to the hordes of hikers near the rim).
Here's the thing: the Kaibabs and the Bright Angel Trails are called the "corridor" for a reason: they get a lot of use, and they are designed for it. The impact of a couple thousand R2R2Rers a year—maybe a few dozen on a couple of busy fall and spring weekend days, most of the year far fewer—pales in comparison to the near-the-rim use and to the mules. Curtailing running is a solution looking for a problem: runners mostly leave before dawn or shortly thereafter, long before the hordes descend below the rim. By the time they return, they're moving slowly uphill and spread out. Most R2R2Rers run when the North Rim is closed, and appreciate the solitude beyond Phantom Ranch.
That's the sort-of 30,000 foot view. But when it comes to managing areas with high backcountry impact, there are two major issues: human waste disposal and trail impact. Over the years I've seen this dealt with in many instances, from hiking the Appalachian Trail to managing backcountry huts in the White Mountains in New Hampshire. And good golly, if you want to see high impact, come to the Whites on a busy weekend where a single trail (in the fragile alpine zone) can see 1000 hikers in a day. Beyond that come things like visual impact, trash removal and the like. But the two big ones are human waste disposal and tread impact.
So back to the Whites. Human waste disposal is a big deal there. Along most of the Appalachian Trail, waste can be handled by dispersal (many shelters down south have designated toilet areas where you dig a hole, do your business, stir the product and cover it and it decomposes) and most others have pit toilets with crude composting systems. When the hole gets filled, the privy is moved, the old waste decomposes, and everything is good. It's moist and generally gets warm enough to decompose the waste relatively quickly (including any TP you bury) and as long as it's not near water sources, you're fine. (In the desert, toilet paper should be carried out, but you need much more education regarding why you should walk along with a bag of poop-covered paper until you reach a trash can.)
This is an example of how dispersal works in many instances. However, in areas with very high use, dispersal becomes a liability. With too many users, dispersal begins to either impact too large an area, or too frequently impacts the same area, leading to problems of overuse. This is when "concentration and management" is the best policy. This is what Phantom Ranch and the campgrounds along the route already do: people are encouraged (by the real facilities where you can dispose of things like toilet paper) to use them, and the waste is managed (by staff and septic systems, although they would probably do better with composting toilets, but that's a whole different discussion). And trail runners? Who creates more waste: a trail runner who is below the rim for 8-16 hours, or an overnight hiker who is down there for two or three days?
(In the White Mountains, use first peaked in the late 1960s and in to the 70s, when "tent cities" sprung up at several sites, including alpine areas near backcountry hut facilities. Before that time, waste was buried in gaboons and tin pits, and each shelter or campsite below treeline had a "goat" for food waste. Now, camping is restricted above treeline and encouraged at established shelters and tentsites which have platforms, hardened pads and managed composting toilets, and huts now compost as much food as possible, have composting toilets [for the most part; above treeline it is too cold to compost waste, so it is stored and flown out] and trails, which ranged up to 30 feet wide in the 1970s, have been lined with "scree walls" to keep people on one narrow section and allow the trail to revegetate. Use has continued to increase; on a busy day above-treeline backcountry areas will see hundreds or perhaps thousands of hikers, mainly due to proximity to population centers nearby. The facilities encourage some of this use, but without them there would be high use and no management. Oh, and while some food is flown in by helicopter, fresh food is packed in—uphill, mind you—on the backs of humans; mules were last used in the '60s)
Concentration and management works for trails as well. The Corridor trails are very well maintained for the high traffic they see, especially from mules. The activity of even a couple thousand R2R2Rers is negligible compared with the daily plodding of the mules. Trails are well built, and again, it comes to concentration and management. If anything, runners should be encouraged to use these hardened, drained trails which are built for use and frequently maintained.
Other impacts: 40 R2R2Rers over 40 miles is one per mile. The visual impact is minimal. R2R2Rers better be conscientious enough to carry out their goddamn gel packs and water bottles. Perhaps the park service could establish an animal-proof "day storage" unit at Phantom Ranch and Cottonwood, where runners could leave supplies for a few hours during the out-and-back part of the hike. Use could require a backcountry tag or permit be attached to a bag containing such food, so that miscreants leaving food behind could be identified (or something of the sort). And education to make sure that no one is littering (although I would bet a lot of the gel packets are fat tourists who suck three down two miles down the trail and throw them by the wayside). There are some cantankerous hikers who bristle at anyone going faster than them on the trail. "You won't have knees when you're my age" they yell, as you swoop on by. Runners should be courteous in passing other users—especially in narrow areas—but shouldn't be singled out for their speed. If we want to get rid of some users, horses and mules, which require other users to vacate the trail completely, should be studied.
Trail running is increasing, and we shouldn't ban it just because people haven't done it as much before. For a lot of runners, the R2R2R is something to train for for months, instead of, say, a marathon or an ultra race. These people are fitter and healthier because of this goal. This should be encouraged. The Park Service, instead of a reactionary policy, should create one which acknowledges that these activities are going to take place, and encourages users who undertake them to be prepared and safe. (This article in Appalachia does a good job of describing the increase in trail running.) Trail runners should be ambassadors for the trails, as well. They should make sure that fellow runners in trouble are helped (or don't get in to trouble in the first place) and that impacts are minimized and avoided. Permits—particularly if they require travel on certain days and times—may well create situations where unprepared runners undertake the trip because they got a permit, or people take trips on days with adverse conditions because they can't wait a day for temperatures to drop or weather to clear. It's reasonable to build trail running in to the Grand Canyon plan. It's illogical to talk about somehow curtailing it all together.