I'd be surprised if unbleached brown paper, screwed up as padding, came out worse environmentally than oil derived plastic. It's probably polythene or variant which doesn't really biodegrade and lasts for centuries. Or leaches chemicals or ends up in the ocean and photodegrades to little bits.
How do we evaluate 1,000 year lifespan vs months for impact? Does that change if we consider it ending up inside wildlife or the ocean foodchain?
Here in my part of the UK at least it's essentially non-recyclable.
I think I'd like to use less, but that's very much an instinctive view. I'd rather corn starch replacements for much packing as it better suits a one time temporary use.