Alternative to `RunSessionHandler` for accessing MCP servers

Now that RunSessionHandler has been marked as experimental, what is the correct way to access the MCP servers in order to send notifications at a later time?

We currently have:

var mcpServers = new ConcurrentDictionary<string, McpServer>();

options.RunSessionHandler = async (_, mcpServer, token) =>
{
    if (mcpServer.SessionId == null)
    {
        // There is no sessionId if serverOptions.Stateless is true.
        await mcpServer.RunAsync(token);
        return;
    }

    try
    {
        mcpServers[mcpServer.SessionId] = mcpServer;
        await mcpServer.RunAsync(token);
    }
    finally
    {
        // This code runs when the session ends.
        mcpServers.TryRemove(mcpServer.SessionId, out var _);
    }
};

We are then able to send notifications to the active servers.

(initially posted here)

Start by reviewing the RunSessionHandler and McpServer APIs, especially McpServer.SessionId and RunAsync, and compare them with the discussion linked from issue 1382. Determine the supported way to retain access to active MCP servers and send later notifications, including how stateless servers should be handled.