Check the IP address assigned to the device via DHCP. Does it match the IP range for the DHCP Network configured in Server.app's (disabled) DHCP service? I.e., is the IP address in the 192.168.1.0/24 range?) Also, try
ifconfig -a on the Mac mini and look for a bridge that contains your WiFi device as a member. In the example below (MAC address obscured), I see that my WiFi device (en1) is in
bridge100. Does the
inet range for the bridge differ from the IP range assigned to your device?
bridge100: flags=8863<UP,BROADCAST,SMART,RUNNING,SIMPLEX,MULTICAST> mtu 1500
inet 192.168.2.1 netmask 0xffffff00 broadcast 192.168.2.255
inet6 fe80::c82a:14ff:fe55:fc64%bridge100 prefixlen 64 scopeid 0xa
id 0:0:0:0:0:0 priority 0 hellotime 0 fwddelay 0
maxage 0 holdcnt 0 proto stp maxaddr 100 timeout 1200
root id 0:0:0:0:0:0 priority 0 ifcost 0 port 0
ipfilter disabled flags 0x2
member: en1 flags=3<LEARNING,DISCOVER>
ifmaxaddr 0 port 5 priority 0 path cost 0
If so, try deleting the Network from the DHCP service in Server.app. This fixed the problem for me. After deleting the Network (I think I had to renew the DHCP lease, too), my devices are now assigned IPs in the range of the bridge.
I think that when Internet Sharing starts, it creates the bridge to support Internet Sharing. So your devices should get IP addresses in its range. I don't know why the Server.app's DHCP Networks interfere with Internet Sharing, since the service is off, but maybe because Internet Sharing forces DHCP on behind the scenes, and the Network settings are picked up regardless. I'm also not sure why an IP in the DHCP service's Network doesn't allow for internet connectivity, but I guess it may have something to do with not matching the bridge's network mask and/or some configuration of
pf that Internet Sharing does behind the scenes.
I hope this helps.