As long as the paper is safe, then a little bit every now and then shouldn't hurt. Sometimes, as a special treat for my budgies, I fill up a bowl with scraps, toy parts, treats, and small willow sticks. The birds absolutely love it, and it distracts them from the bottom of the cage. Usually the taller the cage is, the less likely your bird is to spend time running down there. Have you tried giving your birdie a sea grass mat or a basket of shredding toys? They're easy to find online or in pet stores.
Worst case scenario, you may want to consider putting a grate on the bottom of the cage.
Overall, your budgie's behavior is completely normal. I'd recommend adding some more shredding toys, as that's what your feathered friend seems to be looking for.