Viruses are right at the interface of inanimate matter and life. However, recent experiments [Sakai et al., J. Virol. 92, e01522-17 (2018)0022-538X10.1128/JVI.01522-17] have shown that some influenza strains can actively roll on glycan-covered surfaces. In a previous letter [Ziebert and Kulić, Phys. Rev. Lett. 126, 218101 (2021)0031-900710.1103/PhysRevLett.126.218101] we suggested this to be a form of viral surface metabolism: a collection of spike proteins that attach to and cut the glycans act as a self-organized mechano-chemical motor. Here we study in more depth the physics of the emergent self-rolling states. We give scaling arguments how the motion arises, substantiated by a detailed analytical theory that yields the full torque-angular velocity relation of the self-organized motor. Stochastic Gillespie simulations are used to validate the theory and to quantify stochastic effects like virus detachment and reversals of its direction. Finally, we also cross-check several approximations made previously and show that the proposed mechanism is very robust. All these results point together to the statistical inevitability of viral rolling in the presence of enzymatic activity.