It is not possible to edit secure subpage attributes unless the user has "Project - Edit Management" rights, even if the user has the "Project - Subpage <subpage> - Edit" rights.
Any Clarity release.
This is working as designed in the Classic UX.
It is necessary to have edit rights to the instance before having rights to edit a subpage. If the instance is not editable to begin with, the subpage edit rights will not grant edit rights to any attributes in the instance, even if those are in the secured subpage.
If you grant edit rights to the instance, it would be possible to add the rest of the attributes to secure subpages and use those rights to restrict the instance edit rights instead.
In the Modern UX, check Secure Field Level Access to ensure only specific groups of users can access specific fields or attributes.