So yeah! I don't have anything I can show yet, but I'll talk about some stuff that's relevant here.
George mentioned parametric CAD, and that's an excellent example. I'd also point to video games — the way their physics engines work is often quite similar to a constraint solver, and the "UI" over the solver is the game itself. Some games like World of Goo or the new Legend of Zelda: Tears of the Kingdom expose the constraint system quite directly as a core part of the gameplay. I'd also point to artistic 3D modelling tools, which differ a bit from CAD in that the interface(s) to the constraint system(s) are more focused on creative expression (eg: "make this thing 'look at' that thing") rather than precision (eg: "make the angle between these two lines 90º").
For the part where you're wondering about multiple valid solutions, I think it depends on how continuous vs discrete your domain feels. If it's very continuous, you might want some of the approaches in Bret Victor's essay
Ladder of Abstraction. If it's more discrete, check out the Ink & Switch essay
Untangle.
I hope that helps. Feel free to ask for more examples or more explanation of anything I referenced.