The important thing about gardening is to have a mix of ingredients, not just one that overwhelms all the others. Cardboard is carbon and commercial glues.
However anyone wants to spend time breaking down cardboard is up to them. But applying it to the garden, the soil needs to be mixed in or the cardboard buried. It's the soil critters that break it down, and if it's dry, or not under thick leaf mulch or several inches of soil, they just can't be in that environment.
And what nutrients are being added to the soil by cardboard? 3" deep leaf mulch, maintained at that depth, checked weekly, will suppress weeds, keep moisture in the soil for worms, not just soil microbes, and add nutrients, create the soil food web that we are all looking for.
These days I have no idea what kind of glues they are using to make cardboard, but odds are it's not non-toxic. A lot of cardboard is required to stay in a sterile state for X amount of time, depending on regulations.
Where it doesn't rain in the summer it's a real struggle to keep it wet enough, in any form. The corners curl up, even under a few inches of soil (unless the soil is saturated regularly, which means by hand, which means another chore in the summer) and wind/air gets underneath it, dries it out. I've found it to be very ineffective if I have to provide the moisture.