It doesn't appear that you're reducing your use of boxes, and you aren't reusing the box as a box.
Recycling it as cardboard would seem to be better than using it as mulch, but using it as mulch would seem to be better than putting it in landfill.
By using cardboard as mulch, I'm also not buying wood mulch. Some people do use wood mulch on top of their cardboard to give it a better look, but I just use grass clippings and raked leaves, which people typically also discard. So things that would end up being useless to me instead go back into my own yard.
Now, I suppose one could argue that the grass clippings could be used in a bioreactor to create methane fuel, but that's just more carbon in the air, when really we should be trapping it in soil and plants.
It's also a hell of a lot easier than bagging all that shit and hauling to the street.
I’m not sure if the incremental energy to turn the cardboard into new cardboard (and transport it twice) is more or less than that of growing, mulching and transporting some scrap wood.
Anyway, that is probably the right computation to run.
Where it does get marginal is with all the extra transportation and packaging.
"I then avoid the additional spend in energy, time, dollars, and effort on wood mulch. "
Agreed, that's as good a reason as any. Plus if we get to the point where our biggest problem is recycling or mulching cardboard I'll be happy.
It seems to me you're recycling the nutrients and biological matter back into the soil. This is a 'lower level' recycling than recycling the cardboard back to cardboard, and so less preferable.
Cardboard is excellent substraight for saprophytic mushrooms, which are dominant in certain desirable types of forest soil. It’s a common trick in permaculture circles for simulating second growth forest succession without having to wait forty years (takes about six to twelve month).