Okay, say you're on a 10 Mbps connection. That update takes you what, 5 minutes? Yeah, that's an annoyance, but that's time for a coffee or a bathroom break. Most games don't update daily or anywhere close to it, so your download time is still going to be <1% of your play time for most people.
If your hardware/connection is really awful, yeah, maybe downloading the latest DOOM isn't a great experience, but luckily there's no lack of smaller things to play.
Either way, it's completely irrelevant carbon/energy-wise. Your power usage is going to be dominated by a game trying to render 3D graphics at 60 FPS, not by the tiny increase of the power draw of a router during an update download.
for one thing there is the question of how to attribute carbon emissions for all the "middleboxes" that are drawing tiny amounts of power throughout the whole process but could add up to a lot.
