Yet on a phase diagram sublimation never begins [highlight mine] until well above absolute zero.
First, at (theoretical) absolute zero substances lack the energy to undergo a phase transition. The structure is a perfect lattice and the only energy is zero-point energy.
Second, points on phase diagrams are in principle equilibrium points. Placing a solid in a vacuum creates a non-equilibrium state. Vapor pressure is a value associated with an equilibrium state without beginning or end.
Therefore molecules on the surface of a solid must be above 0 K to have sufficient energy to detach from the lattice. If a vacuum is sustained above a solid then a liquid will not be a minimum free energy state and molecules will tend to escape into the vacuum.