I can understand why a callback module must provide init and handle_call functions. init is for creating the initial state, and handle_call is the main purpose for creating a server process: to serve requests.
But I don't understand why handle_cast is required. Couldn't gen_server module provide a default implementation, like it does for many other callbacks? It could be a noop like
handle_cast(_, State) -> {noreply, State}.
It seems to me that the majority of callback modules provide noops like this one anyway.
handle_cast is similar to handle_call, and is used for asynchronous calls to the gen_server you're running (calls are synchronous). It is handling requests for you, just not with a reply as a call does.
Similarly to gen_call it can alter the state of your gen_server (or leave it as is, up to your needs and implementation). Also it can stop your server, hibernate, etc. just like your calls - see learn you some erlang for examples and a broader explanation.
It "can be a noop" as you said in the question, but in some cases it's better to implement and handle async calls to your server.