[Feature] De-couple Starlette from FastMCP to make it easier to implement MCP endpoints in other frameworks

If someone wanted to implement an MCP server in another framework such as Django or some other ASGI implementation, one would have to use the mcp.server.lowlevel.Server class and implement handlers for the low-level server. This means we have to sacrifice a ton of functionality from FastMCP that really has nothing to do with the python server backend. For example tool/resource/prompt managers, decorators, etc would have to be re-impemented.

It seems that these features of FastMCP could be decoupled in to a base class or mixin that handles all the tool/resource/prompt management & decorators, as well as the integration with the low-level Server. You could then extend that class to implement your own server with all the routing and request/response logic.

The proposed base class would have minimal dependencies on these framework-specific concerns:

Base Class Dependencies:

• MCPServer from mcp.server.lowlevel.server
• Tool/Resource/Prompt managers
• Basic handler methods (list_tools, call_tool, etc.)
• The decorators (@tool, @resource, @prompt)

Server-specific Responsibilities:

• Settings handling (Settings class with HTTP config)
• Transport logic (stdio vs SSE)
• Routing (custom_route, Starlette routing)

class FastMCPBase:
    """Base class or mixin with tool/resource/prompt management and core handlers"""
    
    def __init__(self, name: str, instructions: str = None):
        self._mcp_server = MCPServer(name=name, instructions=instructions)
        self._tool_manager = ToolManager()
        self._resource_manager = ResourceManager()
        self._prompt_manager = PromptManager()
        self._setup_handlers()
    
    # All the @tool, @resource, @prompt decorators
    # All the list_tools, call_tool, etc. handler methods
    # Context management

class FastMCP(FastMCPBase):
    """Starlette-specific implementation"""
    
    def __init__(self, name: str, **settings):
        super().__init__(name)
        self.settings = Settings(**settings)
        self._custom_starlette_routes = []
    
    # Only Starlette-specific methods like sse_app, routing, etc.

The implementation in the other library could look like this:

class DjangoMCP(FastMCPBase):
    """Django-specific implementation"""
    # Django-specific routing & transport logic

One slight complication could be the Context class as it has a reference to the FastMCP instance (self._fastmcp. However the only function that's used is read_resource which would already be on the base class. So possible just need to change the type hint to use the base class.

Review the low-level MCPServer integration, tool/resource/prompt managers, decorators, and handler methods described in the issue. Compare Starlette-specific settings, transport, and routing with the proposed FastMCPBase boundary, including Context's FastMCP reference; done means framework-independent code retains these capabilities while FastMCP keeps Starlette concerns.