And do they exert any control over the software and configuration on it? That was kinda the crux of it after all.

Controlling your hardware without consent that they legally can't ask for would be illegal hacking.

They do however have the right to mandate certain configuration parameters just how they are allowed to mandate you connect something that isn't a noise generator to e.g. a cable TV outlet. Well, being able to limit you to connect devices that conform to some spec.