Companies are not liable if they have proper ID of the person who submitted the content and can provide that to a plaintiff. If they have not made a good-faith effort to know who submitted this info (like taking ID, not just an email address) then they're taking responsibility for the submitted content.
Which means sites that have responsible moderation can still allow anonymous contributions.
The real problem is the inherent asymmetry of legal battles, where the wealthiest can fight forever with endless motions and have near-total impunity while a legal action would basically nuke a normal person's life. Not to mention the fact that an international border can often make this whole conversation moot.
Anonymous contributions, up to the point of somebody compromising the service? With the quantity of password hash thefts, I suspect we'll see even more ID thefts this way.
I can't imagine using any service that asks for ID, except perhaps from the well-established giants, so an exception for identifiability would effectively be a gigantic moat granted to the largest internet companies to keep out competition. Anything like that would need to be paired with massive anti-trust changes, as well as perhaps government take-over of the giants as utilities, none of which sounds very appealing...
That said, don't take any of my rambling as discouragement, your type of thinking is exactly what we need, we need massive amounts of policy discussion and your suggestion is very innovative.
One of my issues is the lack of liability in practice. The poster is technically liable but they're anon, behind proxies, foreign, etc. and unaccountable. It results in people being harmed online without recourse.
These companies should have a duty to know who their users are.