There seems to be little secondary literature on this so answering the OP questions fully would take some serious digging into the original sources. One promising secondary source that I was unable to locate is Ceccarelli and Koetsier, Burmester and Allevi: A Theory and its Application for Mechanism Design at the end of 19th Century, Proceedings of IDETC/CIE 2006 ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. I will only give some comments.
There has been a revival of interest in the old drawing techniques recently due to their digitization, for touch-based input sketch-based techniques give a more natural feel and fluidity than conventional interfaces that employ control point manipulation of splines like Bezier curves. I suspect that the reference to clothoid comes from modern digital reconstructions of French curves, for instance McCrae and Singh stitch them from clothoid pieces. But there is no connection to Burmester's works, only reference to the Lexicon der gesamten Technik, and earlier Singh used cubic NURBS (Non-Uniform Rational B-Splines) to the same end. By the way, Lexicon der gesamten Technik (Encyclopaedia of All Technology and its Auxiliary Sciences) is not a book by Burmester, it is a multi-volume encyclopedia first published by Lueger in 1894–1899 that had a picture of a complete Burmester set, it is unclear that Burmester directly contributed to it.
Burmester's doctoral thesis was on the geometry of isophotes (lines of equal brightness), but his opus magnum Lehrbuch der Kinematik (Textbook of Kinematics, 1888) developed kinematic synthesis of curve-drawing linkages, an elegant and sadly forgotten theory originated by Reuleaux (Gibson has a brief section on it in his Elementary Geometry of Algebraic Curves (1998), which ironically does not even mention Reuleaux). Roughly, the idea was to sketch various curves by designing linkages and other mechanical devices whose points would trace them in motion, such as the classical four-bar linkage employed in Watt's engine. It seems likely that the curves Burmester used were linkage coupler curves (traced by a point on the coupler bar). The cubic of stationary curvature (an asymmetric strophoid, below), traced by the four-bar linkage, is reminiscent of the shapes in French curves. Hartenberg's Kinematic Synthesis of Linkages, published when the subject was still lively (1964), gives some history:
"Modern kinematics had its beginning with Reuleaux. His now classical "Theoretische Kinematik" of 1875 presented many views finding general acceptance then that are current still... Reuleaux regarded a mechanism as a (kinematic) chain of connected links (or parts), one link being fixed... Through hindsight we saw in Watt's straight-line linkage the amorphous beginning of an ordered and advanced synthetic process. Nearly a century later Reuleaux identified synthesis as a concept, as an entity that might be and must be pursued to guide the designer through the maze of mechanisms. His views were limited to only type synthesis: by this is meant the determination of the type of mechanism for a given job... which lies ahead of the related and more recent fields of number and dimensional synthesis.
[...] Geometers and algebraicists of the 1870s became interested in linkages as curve-drawing devices, not as hardware; their work has since been made part of the corpus of the kinematics of mechanisms. It was discovered that a link motion can be found to describe an algebraic curve of any order. In particular, the coupler-point motion of the four-bar linkage (then called a "three-bar motion") was studied. Samuel Roberts showed that a coupler point describes a curve of the sixth order.
[...] Concepts far beyond those of Reuleaux were added to the picture he had attempted to paint; the whole is now called synthesis of linkages... The third and last kind is dimensional synthesis, or the determination of the proportions (lengths) of the links needed to accomplish the specified motion transformation. In Germany, Burmester was in accord with Reuleaux's fundamental concepts and most of his nomenclature and definitions. Making extensive use of mathematical principles (mostly geometrical), and considering displacement, velocity, and acceleration, Burmester's "Lehrbuch der Kinematik" (1888) developed
geometric methods that furthered analysis and showed the way to synthesis.
By the way, Hartenberg uses "analysis and synthesis" in their original ancient sense described by Pappus.
Cubic of stationary curvature traced by a four-bar linkage: