You are still describing a situation where a client connects directly to a server.
At Google, those client connections are probably kept open at a load balancer. The load balancer has its own connections to servers on the other side, and those will be as short-lived as possible, so that the server is available to service the next request the lb sends its way.
If you start talking about server-side push technologies, you are keeping a long-lived connection open from the server, through the load balancer, to the client. That will affect the dynamics of the load balancing - it's not as simple as round-robin or random distribution that short-lived requests allow. Consider, for example, a server that becomes overloaded. With short-lived requests you can just add more hosts to the LB pool. With long-lived requests you might need to start thinking about migrating connections, or forcefully terminating some connections if the server hits some kind of load threshold, as well as adding hosts.
It's not impossible, it just makes it more complicated and less reliable, which is not what you want for something that just updates a score. As others have pointed out, it's probably just fetching a static file that gets updated periodically.