Similar to #944, per #861

Taking #910 as a template, this is my proposed order of work:

• Adding client class side-by-side existing (Adding basic client class in storage. #964) (likely will factor out the pubsub client in core first, see Adding base client class in core. #953)
• Adding factory constructors to client. (Done by the parent class in Adding base client class in core. #953)
• Move methods from storage.api onto the client and remove api module (Remove storage.api module and move methods onto client #973)
• Replace client in places where a connection is used (Replace optional connection with client in storage methods and functions. #984, Replace connection in bucket constructor. #987)
• Drop the _implicit_environ module
• Update regression tests (may just do this progressively over several PRs instead of a brain dump PR like Attempt at implementing clients for Pub/Sub #910 was)
• Update docs (ditto here)