Optimization can work well for interactive CAD usage, because geometric sketches tend to be close to the intended solution, and it's fast. It also has the nice property of stability (i.e., with optimization, small changes in parameters tend not to cause large perturbations in the solution). Doing more gets into what you mentioned, and is called chirality or root identification in the literature. That's much more intense computationally, but could be useful especially if cheaper approaches like optimization failed.
We've started a 2D geometric constraint solver at https://github.com/endoli/fiksi doing the constraint part of this in Rust. We're using it internally and so far it works well, though it's still experimental. More constraints and especially better behavior around failure are needed. The latter will likely entail at least doing more with degree of freedom counting, though there's some of that already.
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