I'll assume by "game servers" you mean "video game binary and asset distribution servers that support game stores like Steam and Epic and others".
When I paid Comcast for 1.5Gbit/s down, Steam would saturate that downlink with most games. I now pay for service that's no less than 100mbit symmetric, but is almost always something like 300->600mbit. Steam can -obviously- saturate that. Amusingly, the Epic Games Store (EGS) client cannot. Why?
Well, as far as I can tell, the problem is that -unlike the Steam client- the EGS client single-threads its downloads and does a lot of CPU-heavy work as part of those downloads. Back when I was running Windows, EGS game downloads absolutely pegged one of my 32 logical CPUs and left a ton of download bandwidth unused. In contrast, Steam sets like eight or sixteen of my logical CPUs at roughly half utilization and absolutely saturates my download bandwidth. So, yeah... if you're talking about downloads from video games stores it might be that whatever client your video game store uses sucks shit.
OTOH, if you're talking about video game servers where people play games they've already installed with each other, unless those servers are squirting mods and other such custom resources at clients on initial connect, game servers usually need like hundreds of kbps at most. They're also often provisioned to trickle those distributed-on-initial-connect custom resources in an often-misguided attempt to not disturb the gameplay of currently-connected clients.
Game downloads, whether on a console or a PC, come from a CDN. The difference is that Steam has a lot of capacity. They can have millions of players all downloading the same game on the same day at gigabit speeds. Console makers invariably cheap out and cannot reach the same level of service.
Hell, it might be the case that console manufacturers are doing the same stupid shit that EGS is doing. Perhaps they wrote their download code back when 50mbit/s was a dreadfully fast download speed for the average USian to have and they haven't updated it since. (And why would they? What's a consumer's alternative other than "Pay 1k or more for a gaming machine that can run games delivered through Steam" or "Don't play video games"?)
