I'm not a Valve employee but I can imagine a rolling update would not be easy considering all these games are constantly connected to the various game servers. They would reconnect immediatelly upon losing the connection and could reach an existing game server running the old software version. Then the rolling upgrade reaches that game server and once again the user is disconnected. And so on.
Blue/green deployments would be better but I'm guessing the scale at which they operate is massive. Back in the old days I managed a fleet of Counter-Strike servers and they were super heavy on CPU/network. Spinning up a complete infrastructure to switch all gamers at once would be too costly, even for Valve.
I doubt anyone at Valve will be at liberty of discussing the specifics you're asking about, so these are my $0.02
