Select Earth in the menu box so that there is now a force of gravity. Now the total energy of the weight/spring/Earth system is the sum of the kinetic energy, the elastic potential energy, and the gravitational potential energy. When is the kinetic energy a maximum?

When the mass is at the equilibrium position. The total potential energy (the gravitational potential energy plus elastic potential energy) is a minimum at the equilibrium position, even though there is some elastic potential energy when the mass is at this location. The kinetic energy is always a maximum when the total potential energy is a minimum (since the total energy is conserved).