How about this option using a 3rd object, Ive had to keyframe it and you would have to further keyframe the visibility of your objects, but you can see the principle of it. I quite like it myself, very flexible set up, not the "flowing" along path style, but more moving along it, Think Im gonna have to go and play with some delay effectors on that now.
Ive used a sphere here but obviously could be another cube.
I also wonder why you don't just align the original Cube primitive to the spline, that would seem to be what you describe. Check the grab for that
Nonetheless i like the transition from 1 object to another.
Deck
press play when open
matrix 3rd object.c4d