The game could become even more interesting if each level of Crown Authority got more differenciated than the others.
The highest level of CA could allow the liege to have a voice in his vassal's succession. According to the research I've done, this was quite frequent in muslim countries and it would make playing as a muslim character, really differentiated.
It could work the same way the pope nominate bishops in catholic countries... or it could take the form of an event where, after the death of your vassal, pretenders go to their liege so that he nominates one. Of course, unhappy pretenders would take action like joining some foreign power
Samely, instead of restraining vassals' ability to start private wars, high CA should only make them ask for the authorization of their liege. Leiges with high diplomatic ability could convince their vassals to back... Leiges with poor diplomatic ability however, would have to endure a relation penalty each time they refuse their vassal's request.
High CA could also allow lieges to exchange titles with their vassals... no more need to revoke a title before giving a new one.
The same way some vassals try to lower CA, some other vassals should try to make it higher. For example, weak vassals that are facing claims of some other strong vassals would realize they need a better protection.