That is what I would expect you to see. Intersection leaves behind only the places where the paths cross each other; because you have a thin, widely spaced path on top you will be left with thin, widely spaced parts of the logo showing through.
If you look at the original tutorial again you'll notice that he uses a thicker path which densely covers the logo, to the point where very little of it shows through. The scribbled path marks the part of the logo that will remain, not the part that is removed.
If you would prefer to scribble the parts to remove, use Path > Difference rather than Intersect, though I think the original tutorial's approach is likely to give a better rubber stamp effect.