Some of the more experienced and affluent members here can probably answer this better (in fact a couple did while I was typing this!), but I'll give a hack at it to the best of my understanding. Any given body of water (or area of land for that matter) can only support a certain amount of biomass before something has to give. For obvious reasons, the addition of some variable (say a small stocking of a non-native species) is going to have a much less drastic effect in a 20,000 acre reservoir than in a 3 acre pond. In an effort to sustain a managable population, most pond owners stock a combination of fish that they know will work well together. This means a predator/prey relationship that already does a pretty good job of maintaining population balance naturally without human intervention. The tried and true combination is LMB/BG, because that's what most people have done for years...probably because the two species don't have too many "special" requirements, and because they are the most popular gamefish and panfish species in the US, respectively. There's certainly nothing wrong with experimenting with other species, and many folks here are trying some unusual things, but there's less guarantee that a non-traditional stocking plan will work over the long term.
BG is the traditionally stocked panfish, but redears (shellcrackers) and pumpkinseeds seem to work pretty well as an additional species; many folks include them as they are beneficial for snail control. Others are less regarded in the pond world. Among these are the green sunfish you mentioned. The biggest problems with GS are that they have a propensity for heavy spawns, and that they have much larger mouths for their size than most sunfish. They can be fairly predatory and compete with other species for forage. Also, due to their spawining habits, they tend to overpopulate and stunt in growth from the resulting lack of food.
Crappie are generally avoided for the same reasons as GS, on an even higher level. They are primarily piscovorous (esp. white crappie) meaning that they compete heavily with bass or other gamefish. They also spawn erratically...they might not produce a good year class for a few years, but when they do, it's a big one - overpopulation = stunting. And because they spawn earlier in the year than bass, they tend to predate heavily on the new bass fry, hurting bass recruitment.
SMB and spotted bass are avoided mostly because their habitat requirements are difficult to provide in a small pond.
The "temperate basses" are mostly avoided, except for the striped bass x white bass hybrid. Several here have had success with the HSB on a put & take basis. They won't generally spawn in a small pond, so they have to be restocked as they're removed. Numbers seem to have to be kept low, as they compete directly with LMB for forage.
Blue and flathead cats are avoided mainly b/c they have a propensity to get really big, even in small waters. One big catfish can eat a lot, leading to problems maintaining the forage base. Flatheads are particularly problematic, because as they get larger they begin to feed exclusively on live fish. Bullheads, on the other hand, are a problem because they tend to both overpopulate and muddy the water. Common carp are avoided for the same water-muddying habits as bullheads. Grass carp (triploid/sterile variety) are often used for weed control, but the numbers are kept low generally...easier to add them if necessary than to take them out.
Toothy species are generally too effective as predators to maintain a forage base with. This imcludes the pike family and the pike-perch family. Also, pike-perch (walleye, sauger, saugeye) don't spawn very successfully in ponds. Pike and muskie require colder water temps than can be provided in the southern part of the country. Chain pickerel might work, I guess (?), in a larger pond with low numbers of the pickerel. I don't know of anyone that's done it, but I'm considering trying to get a few grass/redfin pickerel for a small pond, just to see how they do. They don't get more than a pound or so, might be a fun experiment.
Sorry to be long-winded, but you asked for it!
I'm sure I left some things out (thankfully, I'm sure), but hopefully that'll give you the gist of why certain species are avoided in ponds. It's all about management strategy, and doing things that are highly likely to work over the long term.
