The router in the customer's home cannot be trusted. With cable at least, you are able to bring in your own modem and router. Even if not, swapping it is easy, you just have to clone the original modem's MAC. In practice this is probably quite common to save money if nothing else (cable box rental is $10+/mo).
Note that spoofing source IPs is only needed by the attacker in an amplification attack, not for the amplyfing devices and not for a "direct" botnet DDOS.
I would in fact guess that it's not common at all. Setting up your own cable modem and router is going to be intimidating for the average consumer, and the ISP's answer to any problems is going to be "use our box instead" and they don't want to be on their own that way. I don't know anyone outside of people who work in IT who runs their own home router, and even many of them just prefer to let the ISP take care of it.