According to the systemd documentation, UseDomains only affects systems that get their network setup from DHCP, which is not the case at my site. Furthermore, it is documented as a way to add a DNS search domain, which would not help resolve hosts that have no domains.
So far, the only way I have found to fix the breakage is to manually disable systemd-resolve and configure a resolver that actually works properly.
According to the systemd documentation, UseDomains only affects systems that get their network setup from DHCP, which is not the case at my site. Furthermore, it is documented as a way to add a DNS search domain, which would not help resolve hosts that have no domains.
So far, the only way I have found to fix the breakage is to manually disable systemd-resolve and configure a resolver that actually works properly.