Can large scale services use a public cloud managed database service?
Yes, it is certainly possible, but it entirely depends on the architectural design of the entire system. Separating application services into manageable components (i.e. microservices) is one approach to this. While Netflix as a whole is one application in terms of the user experience, it is comprised of many applications/services that provide a small fraction of the functionality needed to run Netflix as a whole.
In terms of a database, the same principal can be applied. Netflix doesn't manage one database for the entire application, each component utilizes it's own scalable database service.