I use that natural eco-friendly paper bedding for the rabbits because they like the comfort of it, but no way would I use it under the bird cage grates unless I wanted to drive myself insane cleaning it up.
For Rabbits, Guinea Pigs, Rats, Mice, Ferrets, etc., any type of rodent/small animal that really does REQUIRE a "bedding" that will absorb the urine/feces, the
paper beddings, such as CareFresh, are definitely the way you want to go! A lot of people (including myself when I was a little kid and had hamsters and mice as pets) don't realize that while "beddings" made of different types of wood work very well as far as absorbing the urine and also for covering-up odors,
most wood-based "beddings" or "chips" are at the least irritating to the animals, and at the worst they can make them extremely ill,
or even worse.
Whether we're talking about your regular old Pine wood-chip bedding, or the more-fragrant Cedar bedding, or "Aspen" bedding, any and all of these can cause a slew of health issues to the animals living in them. We have to realize that pets like Rabbits, Ferrets, Guinea Pigs, Rats, Mice, Hamsters, etc. are
literally living directly IN the bedding their are in, with it constantly touching their skin, constantly getting into their eyes, their ears, and up their noses, and they are breathing in the fumes from these beddings all day long,
every day, without hardly a break. Eye irritations/infections, nose/nostril/throat/mouth irritations and infection, severe skin irritations, burns, and infections, and most-seriously they are constantly breathing-in those fumes 24/7, which can cause severe irritation of their Respiratory System, including their lungs, and cause everything from URI's and Lung infections to potentially-fatal Fungal Infections...Plus these fragrant wood beddings are just extremely irritating and cause burns to their skin, their eyes, and their airways.
CareFresh and other similar "paper" beddings are fantastic, they work just as well if not better than the wood beddings for absorbing urine and feces, and they are completely safe, cause no irritations or burns, and are soft and easy on their skin. Yes, they are more expensive than the cheap Cedar or Pine beddings, but that's what you take-on when you bring home a pet. And CareFresh/paper bedding comes in a much cheaper Generice/Store-Brand that is close to the same price as the Pine and Cedar/Aspen beddings. Never buy your CareFresh/Paper Beddings at any pet stores, as you'll be spending an arm and a leg. Instead you should buy them at farm supply stores and at stores like Walmart, all of which sell both CareFresh and their own Generic Store-Brands very cheaply...Incidentally, I have used the CareFresh/Paper Beddings inside of my Brooders for the baby parrots I bred, and it is perfectly safe to put a 2 week-old baby bird in. I got a huge, compacted/compressed cube of Generic CareFresh at Tractor Supply for under $5 (it was $6 less than the same size at Walmart actually)...