Naturally DHCP is handled by the original network when the original network can be seen, yes!

But if the original network is unavailable - perhaps the SSID has changed for example - you need to be able to get to the repeater's configuration page somehow. And the user can't google their problems or download any configuration tools while their connection is down, so you'd better make it simple. So many devices like this will be transparent under normal operation, but if there's a network problem will start their own DHCP server.