My biggest problem with the "figures are necessary for skaters to have great edging" is ice dancing. Ice dancers generally have better edging than any other discipline, yet ice dancing has never involved figures, neither in competition nor training.
Also, others who speak the languages can correct me if I'm wrong, but it's not called "figure skating" all over the world. In certain European languagues, the English equivalent is "artistic skating."
However, for the individual skater--and it seems people tend to focus on ladies singles whenever this topic comes up--I believe that whether or not practicing figures helps edge quality depends on both the individual skater and the coach. Some skaters with some coaches can get edge quality galore by having edging emphasized at every point in the training process, and especially in the actual moves they use in their programs. These individuals respond to specific training. Other skaters need separate time to practice figures in order to correct certain problems, as SkatePixie described. This may either be because these skaters need to break things down before they can incorporate them into more complex skating moves or because the coach does not emphasize edging during the rest of the training process.
Overall, I don't think that the lack of practicing figures has resulted in a generation of clunky skaters. There have been plenty of clunky skaters as long as the sport has been around. But prior to the triple jump era, at least for ladies, edging and flow was a big deal because the judges made it a big deal. Now the most dangerous and high risk element is the triple jump, hence that's what the judges place the most importance on. Young skaters such as Carolina Kostner. to name just one, have superb edge quality.
Also, I don't think it's appropriate to compare skaters from the doubles era, such as Dorothy Hamill, to today's skaters who have to work toward seven-triple programs and 3/3s. Trying to perfect triple jumps takes time and energy. Skaters from the doubles era had more time to work on edging and they knew they would be rewarded for doing it well. Not so today. Doing figures may help some skaters with their triple jumps; for others it may do nothing.
However, where I think it is appropriate to compare is that short period of time when for ladies, five and six triple programs were the rule and Olympic programs generally had one 3/3 AND the skaters had been trained in and/or had competed in figures. Depending on the skater, that period is mostly limited to the early '90s.
Rgirl