Hello Dominik Almus
If the DHCP service works after restarting but then stops assigning leases again, there might be an issue with the DHCP service itself or its dependencies.
Check the "Dependencies" tab in DHCP Server service, does it include all necessary services and drives? does it contain AD DS?
DHCP relies on the DC for name resolution (DNS) and sometimes for authentication, DC service should start before the DHCP service.
Thanks,
Karlie Weng