An analytical framework is proposed to describe the elasticity, viscosity and fragility of metallic glasses in relation to their atomic-level structure and the effective interatomic interaction. The bottom-up approach starts with forming an effective Ashcroft-Born-Mayer interatomic potential based on Boltzmann inversion of the radial distribution function g(r) and on fitting the short-range part of $g(r)$ by means of a simple power-law approximation. The power exponent $\lambda$ represents a global repulsion steepness parameter. A scaling relation between atomic connectivity and packing fraction $Z\sim \varphi 1+\lambda$ is derived. This relation is then implemented in a lattice-dynamical model for the high-frequency shear modulus where the attractive anharmonic part of the effective interaction is taken into account through the thermal expansion coefficient which maps the $\varphi$-dependence into a $T$-dependence. The shear modulus as a function of temperature calculated in this way is then used within the cooperative shear model of the glass transition to yield the viscosity of the supercooled melt as a double-exponential function of $T$ across the entire Angell plot. The model, which has only one adjustable parameter (the characteristic atomic volume for high-frequency cage deformation) is tested against new experimental data of ZrCu alloys and provides an excellent one-parameter description of the viscosity down to the glass transition temperature.