In the course of the conversation I had with them, they said I'd need to use their router so they could run some further checks. I'm sure they only wanted me to tell them which lights were on, but is there any further checks that an ISP can do against the router they provide?
Any number of them. They can even push firmware updates.
Now whether such access is actually worthwhile in diagnosing a fault is a different thing. I've always been sceptical of the idea that ISP support staff would be capable of gleaning anything useful that way. Most faults are electrical in nature and dealt with by openreach who have no interest in your router. The kinds of faults where router access would be useful are the kind of things that ISP support staff are unlikely to want to get involved in like port forwarding.