Even if you control your router/modem, they still control the other end, it connects to. So i think it depends a bit where you’re going with this. If you’re worried about them doing packet inspection, or logging IP numbers you connect to, I don’t think there’s a big difference. They could do it anywhere.

A router interfaces with your local network, though. So in theory a router can be used to connect to your internal devices and computers and maybe they have an open network share without password protection or something like that. But we’re talking violating your constitutional rights here. It’s highly illegal in most jurisdictions to enter your home and go through your stuff.

I’ll buy my own router because I can then configure it to my liking. And my ISP charges way too much for renting one. And what I also do is not use my ISP’s DNS service. That’d just send every domain name I open to their logfiles. Instead I use one from OpenNIC