Thank you for the post.
If FileMaker Pro on the FileMaker Server is able to open remote and see the files, but other computers on the same network are not, then port 5003 is blocked to the client machines.
As a test, turn off all the firewalls on the host and client machines. If this works and the firewalls need to be on, then configure the firewalls to allow ingoing and outgoing traffic for TCP port 5003.
If turning off the firewalls does not work, then review the installed programs for any security or antivirus that could block the port.