I think your history isn't quite right here. The first widely-used CAD systems used Bezier curves, where the end points of each section do lie on the curve. For cubic Bezier curves, the intermediate points also lie on the tangent lines at the end of each segment. This corresponds quite closely to the constraints that are implied by using a physical drafting spline as in your image.
Bezier curves are still used - for example to define the character shapes (glyphs) in digital fonts.
The next step was the realization that there are other mathematical ways to express the same curve shapes as Bezier curves, which are more convenient for some purposes. A commonly used alternative is B-splines, which can represent everything that Bezier curves can do, plus the capability of having discontinuities of slope (i.e. "kinks") along the length of the curve.
This is valuable in CAD software because there is no difference in the math required to handle a polygon made of straight line segments, or a smooth curve.
But B-splines were not the complete solution to the problem of computer geometry, because they can only represent approximations to circles, and engineers just love circles!
Nurbs (Non-uniform Rational B-splines) were the next level of generalization. The key feature of Nurbs is that they can represent curves such as circles and ellipses exactly. The math required to work with Nurbs is more complicated than for B-splines, but you can write "real-world" CAD software that uses Nurbs to define every geometrical object. Since there are now well-established software libraries to do the math that underlines Nurbs geometry, you don't need to understand the math in detail to be able to use it in developing new CAD application software.
International standards for defining and transferring CAD geometry between different computer systems, like IGES and STEP, use Nurbs geometry as their basic mathematical foundation.
- Bezier curves (or their equivalent, Bernstein polynomials): first defined in 1912, first use in computer graphics in the 1960s.
- B-splines: The math was actually discovered in the 19th century, but efficient computer algorithms date from the 1970s.
- Nurbs - first computer implementation in 1989, first practical CAD systems in the 1990s.
Other types of splines have been proposed, including "minimum-energy splines" which model the behaviour of an idealized physical spline as illustrated by the OP), "minimum-curvature-variation splines", B-splines with an extra "tension" parameter to allow the user to control the amount of curvature at different parts of the curve, etc.
In reality, there is no universally accepted objective criterion for judging the "best" spline. This http://levien.com/phd/LevienSequinCAD09_014.pdf discusses some of the issues, and includes the results of a survey in which the minimum-energy spline fit was not subjectively judged to be the "best" fitted curve.
Since Nurbs or B-splines can approximate any spline shape to any required degree of accuracy, the debate about what is the "best" shape is perhaps not of critical importance - you can learn how to use whichever type of spline your CAD system provides to produce the result that you (personally and subjectively) like best. An important practical issue is the "stability" of the spline fit, i.e. whether a small changes in the control points can produce undesirable large and non-local changes in the spline shape.