My question is whether one is able to distinguish different groups of substances (possibly depending on their molecular composition or other factors) depending on their thermal behavior when oversaturated.
Briefly no. Going into more detail, you can with advanced knowledge of chemistry, but we can't teach you PhD level chemistry one posting at a time. This is the sort of thing advanced theoretical chemists do.
You have a fine example from your chart: Na
2HAsO
4 had the most solubility change over temperature. Maybe there's better ones, but you'll find that faster in a reference book than in a mountain of theoretical calculations.
The big problem that I have is that I think your premise is flawed in two ways. You keep mentioning supersaturation. There's no guarantee that any substance will allow that. Again, advanced crystal theory will tell you when and why, but I just know of a couple of examples as trivia, the phenomena isn't ubiquitous. If as soon as the temperature drops, you get the thermal energy back gradually, you can't use that to do work.
Furthermore, you're convinced that you will get all the thermal energy back (I'm guessing you know to expect inefficiency losses.) This is not absolutely true. If the entropy of the crystal forming or dissolving is lower, you can waste some thermal energy that way. There's the example of things with a negative heat of solution: they dissolve better when they're warmed, yet they cool the solution as they dissolve. This effect may be less for some reagent or another, but its still a way to lose thermal energy, while keeping overall energy the same.
Now, phase change, that's a transition that's as straightforward as can be. But we do that all the time. We use melting ice to keep things cool in shipment, or steam to drive turbines. But I'm getting the feeling you simply find that dull. Still, I don't know if you can get the heat back from a solution.