Why do DHCP servers have static IP addresses

I'm not entirely sure bt I think it's because they set themselves as the default gateway and usually the DNS server for whatever devices connect to them, and if their IP changes, the device will have to obtain new information from the router.

That would be my guess as to why routers (which are basicallyt DHCP servers) have to have static IP addresses.

A DHCP server must have a Static IP Address. This means that the system running the DHCP server may not be configured to use DHCP to configure its own TCP/IP stack.