The Dutch astronomer who showed that the Milky Way rotates differentially was Jan Oort. His work demonstrated that stars at different distances from the Galactic center do not orbit with one shared angular speed.
In 1927, Oort developed observational tests based on the motions of nearby stars. His analysis confirmed the central idea proposed by Swedish astronomer Bertil Lindblad: the Galaxy behaves more like a rotating disk with varying orbital speeds than like a rigid wheel.
Oort’s result also helped overturn the older Kapteyn model, which placed the Sun near the center of a seemingly symmetrical stellar system. Oort inferred that the Sun lies tens of thousands of light-years from the Galactic center and takes roughly 225 million years to complete one orbit.
Bertil Lindblad is the main historical near-miss in this question. Lindblad proposed the theoretical picture, while Oort supplied influential observational confirmation and a practical mathematical description for the solar neighborhood. Those formulas introduced the quantities now called the Oort constants.