I use kimai. It’s gotten much more stable over the past couple years and there is a mobile app. (I think it’s a couple bucks now, but works pretty well)
It runs in docker so it’s pretty easy to set up.
I have one customer who is pretty nuts about bills and this is the only way to track and invoice all that nonsense.
First of all, the ISP controls cable modem firmware. They have all the settings and manage the device. You don’t get much control there.
As for your question, I’d say no, for 2 reasons. First, designing that capability is expensive and modems are built for cheap reliability. Second, any hardware to spy is more useful installed in a data center accessible to their user base. There is not much point installing unnecessary tech to one endpoint.
As for router, they are beefier CPU-wise. AT&T has in the past prevented users from changing DNS settings and that could lead to lots of tasty data. Deep packet inspection is becoming more prevalent in home routers as is integration with other technologies. (EERO devices for example).
Make sure to fire up a VPN or something when you need.