Usually the router in a home consumer level piece of kit will act as the DHCP server. I use my Deco in AP mode and it does have a "smart" DHCP service that tries to offer IP addresses when the main DHCP server is unreachable but I disabled that because it was very unreliable.
A quick way to check if the Deco will work while in router mode is to try a dig at the router's IP address and see if there is any response. If there is no response then there is no way to access the routers hostname <> IP information. A timeout error would tell you there is no DNS server exposed at all and a NXDOMAIN response would tell you there is an exposed server but that it will not tell you the information you are looking for.
Pi-hole will function as the DHCP server and many people have it set up in that way when they have a router that will not work for local hostname resolution or their router is not very configurable. You would need to be able to disable the DHCP server on the router for that kind of setup since you only want one DHCP server on any network segment. (There are exceptions to that rule but that gets in to more complex network models.)
You are correct that Pi-hole will not work as the router since it only sees DNS (and in some cases DHCP) information and can not route packets out to the WAN.