They shouldn't be able to tell if you're using their router or not
Don't think you understand. While I agree, they shouldn't be able to tell, it doesn't matter if they can, or can't. They are charging anyway.
why not just use their router and wire yours to it?
I can't answer that for the OP but I know the network equipment my ISP provides is barely middle of the road quality. Plus, it is a "residential gateway" device - meaning the modem, router, WAP (wireless access point) and 4-port switch are all in one box. And, they don't just let us upgrade to something newer just because we want to.
Also, like many ISPs, they offer nationwide access so if you are traveling you can still use your internet access. How? By ensuring any of their customers can access any other's wireless access point - anywhere in the country. So if a customer from Florida, for example, is visiting my neighbor here in Nebraska, if they can see my wireless network, they can automatically connect and use it to do whatever they would normally do if still in Florida.
And while this traveling account is NOT supposed to use up any of my bandwidth, and it is supposed to be totally isolated and firewalled from my own wireless network, do you trust your ISP to keep it that way? I don't
For those reasons, I use my own wireless router and my own cable modem which I know are better than anything my ISP provides. That way, if I want to upgrade just my router, I can. If I want to upgrade just my modem, I can.
Last, and perhaps most importantly, I know there are no "backdoors" in my devices the ISP can use to gain access without my permission or knowledge, nor can visitors from Florida use my router (or my electricity) to do who knows what?
So no. They can keep their router, thank you.
BTW, different jurisdictions around the world have different laws and regulations. My ISP is not allow to charge lease fees for Internet access unless the customer is indeed, leasing the equipment. But that's here, in the USA. And even then, there are some exceptions.
I have to use their cable box for my cable TV/DVR services, for example, and pay their rental fees. No choice.

Well... I can choose one box that lets me record 2 channels while watching a 3rd channel for some ridiculously overpriced rental fee. Or I can choose a deluxe box that lets me record 10 channels while watching a third for an outrageously ridiculous, overpriced fee.
