The Protected version allows other users to create child versions and edit them, but they cannot (as you have discovered) post their edits back.
The owner of the protected parent version can post the edits back. If the child versions are set to public, then the owner (administrator) can change to those child versions and post those versions back into the protected version.
Take a look at Version creation example - Creating one new version:
All employees can connect to both versions. Only the geodatabase
administrator can edit data when connected to the DEFAULT version and
post to the DEFAULT version. When employees connect through the Base
version, they can edit any datasets to which they have been granted
the required dataset permissions.
Also Using the new version:
When Maxine finishes a set of edits, the geodatabase administrator checks Maxine's edits in the Base version. If the changes are okay, they are reconciled with the DEFAULT version to bring any changes that may have been made into the DEFAULT version. Since only Maxine should be editing this data, there shouldn't be any conflicts found during reconciling. At this time, the geodatabase administrator posts the changes to the DEFAULT version.
(emphasis added by me)