Dfariswheel, good summary! I used a duckbill choke in South Africa for a while during the 1980's, when they were very popular on riot- and HD-type shotguns. They did, indeed, require a higher pellet count to be effective. They were available in two "standard" types: 2-to-1 and 4-to-1 spread. Each "flattened" the shot pattern so that it was, respectively, twice as wide, or four times as wide, as it was high (sort of a rectangular pattern).
They were very effective against moving targets, as the flat "band" of shot made it easier to catch the target with at least part of the pattern. However, you were virtually guaranteed to miss the target with some other part of the pattern, which meant that anything in the line of fire, beyond the target, got some too... which led to screams of outrage if one's partner(s) were beyond the target!
Another major drawback was that the duckbill choke just couldn't handle slugs... The 2-to-1 spread choke was claimed to be able to handle slugs, but in practice it launched itself downrange with the slug stuck firmly between its lips, just as its narrower 4-to-1 spread cousin did! If one wanted the longer-range capabilities of slugs, this meant that one had to have a second shotgun available: and if one forgot about this, and mistakenly loaded a slug into the magazine along with some buckshot (a mistake made rather more than once, in my experience, although fortunately not by me!), this would disassemble the choke from the shotgun barrel very effectively.