I mean, if you look at the ALL of the factors involved, one could easily fall into a situation where an angel decimates their corals. My theory (so far, based on everyone's feedback and my own experience/research) is if one overlooks just ONE of these factors, it could lead to getting the fish trap out. But if the steps are followed, I think that a dwarf angel (and maybe other angels) might be some day considered safe to put in most (if not all) types of reef tanks.
First, is the size/type of the angel (which would infer age and possibly origin). As with most fish, typically the younger individuals (to a certain extent) have an easier time acclimating to captivity than do the older ones, which might mean that the younger individuals are more likely to quickly accept a wide variety of prepared foods and less likely to revert to picking since they are getting their dietary needs met. Also, as some have already pointed out, there are certain angels that seem to do better in captivity (meaning they accept prepared foods quicker and are easier to satisfy their dietary needs).
Second, there is quarantining the fish. I know a lot of people out there don't quarantine their fish, but I think when you're introducing an angel into an established reef tank with several hundreds/thousands dollars worth of corals, it might be worth the investment to not only make sure they are healthy/free of disease, but also to make sure that your angel is eating a variety of foods without the temptation of picking at some of your prized corals. This actually might be the determining factor on whether you are successful going forward. I personally wouldn't add the angel to the DT before he/she is eating EVERYTHING I want them to eat when they are in the DT.
This leads us to the food being provided and the frequency... although just flakes and pellets might be fine, I would think that a mixture of Formula 2 flakes/pellets, frozen preparations, and nori, would likely satisfy their dietary needs. As important, however, is probably the frequency of feeding. If possible, I think the optimal feeding would be a clip/rock with nori attached in the morning (so they have something to pick on when they "wake up"), maybe a small feeding from an automatic feeder of flakes/pellets come early afternoon, and then a mixture of frozen preparations when you get home from work in the evening, and depending on what your light schedule is, maybe another small feeding before you go to bed. And, as saix88 just pointed out, the feeding needs to remain consistent and provided long term, not AFTER you've noticed that the angel is already picking at the coral. As some have pointed out before, I think once the coral is in distress (which typically creates more mucous), the excess mucous provided by the coral becomes too much of a temptation for the angel, and they start to pick at it. Also, this type of feeding obviously would require a good skimmer and other filtration (DSB, LR, Phosphate reactor, etc.).
This then leads to the size of the tank (and, indirectly, the amount of LR and exposed "picking surface" you have). The smaller the "picking surface" the more frequent feedings probably need to be. For instance, the above described feeding schedule would either be for a tank with little "picking surface" or a tank with little algae growth. The more algae growth or picking surface, the less frequent the feedings must be (probably no less than one feeding per day though).
Last is the other animals in the tank. If there are competitors of the same food (i.e., algae), then you would think that the angel is more likely to look for other food sources (your corals). This, of course, much like the "lack of picking surface" situation described above, should be able to be remedied by providing more frequent feedings of the right foods.
Something that I'm still unsure of is whether they seem to pick at a particular kind of coral, or corals of a particular color. Open brains, zoas, and some leathers seem to be a commonly picked on corals, and probably any other "fleshy" coral that tends to exude a good amount of mucous, but it doesn't seem that color plays a factor (or at least based on the relatively small amount of information I'm dealing with so far). Even still, I think if introduced in the right way and provided with a sufficient diet, that these corals should also be safe.
So, up to this point, I would be interested to see if someone that has taken all (or most) of the steps above still had a problem with their angel, and if you didn't do one/a few of the steps, which one(s). I'm not saying that there are not exceptions, and that people can't "get away" with doing less, but I'm trying to find a process that will almost guarantee success with this beautiful fish in a reef tank, and my guess is that the process described above might just do the trick.
Keep the feedback coming guys (the larger the data pool, the better)... anyone out there do the steps described above and STILL have had troubles with their angel in their reef???