In Scotland I was with Telewest, then Virgin, and my memory is always that the DHCP pushed out the external IP of the ISP's DNS servers.

Nowadays I'm in Finland and definitely the router runs no DNS service, the DHCP service advertises the ISP resolvers.

Probably depends on the region/ISP I guess, but I had no expectation that it would be the more common option.

American here, most of ISPs here do it as well. With modern router hardware, there is plenty of hardware available to run tiny DNS server that caches and forwards all requests to ISP upstream. Memory overhead is probably about 50MB and CPU overhead is trivial, probably .1% or less.