The Gelfand-Tsetlin subalgebra S of U(gl_n) is an important structure in Lie theory which remains underexploited decades after its discovery.  Viewing U(gl_n) as a Coulomb branch gives a surprising new perspective on this algebra which is strongly adapted to S and sheds considerable insight on the Gelfand-Tsetlin modules over gl_n, those in which S acts locally finitely (for example, allowing us to classify the simple modules for the first time).  On the other hand, techniques developed to study G-T modules can be profitably applied to other Coulomb branches.