You feel how you feel, but it seems misdirected and weirdly focused on recycling over other options. Many people reuse paper either for other projects or other gift giving and then there are those of us who just don't use wrapping paper at all.
Many years ago, after being fed up with wrapping paper (which like others, my area doesn't recycle and I just found a pain using, and with multiple nonreading kids there kept being confusion even if I picked different wrapping paper for each child), I bought one of these jumbo storage bags for each of my kids with the ones old enough picking out their own.
Every birthday, Children's Day and New Year, their presents go in, we bring them down, we have a countdown to open and they pull things out at their own speed. When little, they loved having a really big bag and doing it all themselves, now they love hearing us bring it down (the material is not quiet), I love that we have years of photos of them with their bags, that there is little mess after to deal with and little stress in maintaining - right now, the bags are sitting on my dresser and anything I get in for them for New Year, I just put it straight in their bag. When it's not close to a gift giving time, they're flattened and hung up. That plus keeping around gift bags to pass along, it makes it really easy.
As others said, the order is reduce, reuse, recycle, with some adding some extra steps before recycling, but it's always the last thing to consider in this.