One interesting and sort of unhappy artifact of CAD is the adoption of B-splines and NURBS as the primal basis for modeling. The whole point of B-splines is that they are the obvious basis for maximally continuous splines of a certain degree (i.e., degree n gives C^{n-1}). This is much more than G continuity. But in CAD, it's often the case that all you care about is just G continuity.
So you run into a weird situation where CAD software may pass around NURBS or B-splines with multiply inserted (or even fully inserted) knots, seriously reducing the need for using splines in the first place.
The problem is that splines are a really inconvenient and even unstable basis for doing numerical work... which is what all of CAD is.
Curious, is there some alternative that would give you both higher order continuity and numerical stability? Or are they fundamentally at odds?
IMO, higher order continuity is a red herring. You can make something approximately high order continuous (say, to 10+ digits, or whatever you like) piecewise much more easily than enforcing mathematically exact high order continuity. Once you think of continuity as something to achieve approximately, standard methods from classical approximation theory suffice.