I'm not a lawyer. But IIUC, they probably care because of issues like contributory or vicarious copyright infringement, at least in the US.
If they are providing you a service through which you infringe someone's copyright (by playing a video game in violation of its licensing agreement) in exchange for giving them money, then they are risking a lawsuit. If they knowingly enable infringement (e.g. by rolling out a gaming service that lists and can automatically install games where they know, or could easily have checked, that the terms do not allow you to install on such a service), then they probably run a higher risk than if they don't knowingly enable it (e.g. by letting you rent cloud machines with GPUs, completely blind to what you're running on them), but in both cases they run a risk.
I'm sure there's a lot of law, and case law, that sets boundaries for this, but I could easily imagine cloud gaming falling on the wrong side of the boundaries.
If they are providing you a service through which you infringe someone's copyright (by playing a video game in violation of its licensing agreement) in exchange for giving them money, then they are risking a lawsuit. If they knowingly enable infringement (e.g. by rolling out a gaming service that lists and can automatically install games where they know, or could easily have checked, that the terms do not allow you to install on such a service), then they probably run a higher risk than if they don't knowingly enable it (e.g. by letting you rent cloud machines with GPUs, completely blind to what you're running on them), but in both cases they run a risk.
I'm sure there's a lot of law, and case law, that sets boundaries for this, but I could easily imagine cloud gaming falling on the wrong side of the boundaries.