They require a signature from the owner of the copyright when submitting a copyright complaint. Do you think they should require a signature from the owner of the copyright when submitting content?

They have no way of even knowing who the copyright holder is barring copyright registration which is optional and uncommon, let alone who actually has permission from that person. There are already copyright penalties if someone wrongly claims to have permission. That's why it works the other way: because trying to do things backwards imposes impossible demands on people trying to follow the law.

EDIT: By way of example, there's substantial controversy over who owns a copyright as famous as Happy Birthday.