Hello @Renfield-64!
My name is Virginia and I work with the LanSchool Support Team. I'm sorry for the issues that you are experiencing. However, I'll be more than happy to help.
The previous support person that helped you, is correct. When we see issues with the teachers not showing up in the tech console, it's because they are using a LCS. I understand that you are not using LCS and that leads us to believe that there is a routing issue or something potentially filtering/blocking them. If the teachers and tech console are on a different SSID, you will need to make sure the ports are open for that communication. This is why we suggest tracing software such as Wireshark.
One thing you may want to try before tracing software is to use either of the broadcasting methods: IP Broadcast or IP Multicast. If your network has multicast routing enabled, you can discover them through that. If not, you will need to be on the same subnet as the staff devices.
Please let me know how this goes.