God created man to be unified with the Spirit, and so our soul has the potential to be spiritual, and thus is reflected in how we can, by our own works, manipulate our soul to imitate spiritual effects.
The Spirit is the true source of the spiritual life and its spiritual effects, and without the Spirit and his grace, these spiritual effects are dead, aggregates within a being with spiritual potential cut of from its source of spiritual life.
After the fall, Adam was doomed to die like a pine tree cut from its roots is doomed to die: it doesn’t die right away, but without its root, without the life which brought all of itself together and nourished it, fades. The clay was breathed to life by the Spirit, but without the Spirit, although Adam might have had still had many spiritual effects that arose from that Grace, nevertheless without the Spiritual, all of them will decay back into the dust. And this is what the “spirituality” that Mr. Harris speaks of, or the Hindu you allude to: an inner life that manipulates spiritual effects, creating an illusion of life, but in reality these effects and experiences are merely the pieces of a dying corpse that just happens to be together, for a moment, before fading into dust. They have no life, they are just the leftovers of the spirituality of a dead man. Vanity of vanities, all is vanity and vexation of the spirit.
Just as the flower in the vase works in vain to sustain itself, this “spirituality” is just at the effects of spiritual life cut off from its source, an artifact, falling apart as we speak, an attempt to do what only the Spirit of God can do.
However, through the internalization of the grace given under the symbol of baptism, all of these dead works can be resurrected in Christ, if only we work with the Spirit to order these effects in him and for him. This is why God accepted the works of Cornelius the Centurion, even if they were not formed by the Spirit, if Cornelius believed in Christ and brought these works up as an offering to be made living by the Spirit of God.
Just as the body without a soul is just an aggregate of parts and forms that just so happen to be next to each other, even if it looks like a living body for a time, an inner life without the Spirit of God is just an aggregate of fading spiritual effects. But just as material parts, although dead by themselves, can become living in being integrated within a soul being born, these spiritual approximations that Harris speaks of can be a part of being born again in the Spirit.
Or something like that.