Brother Fax T78 wont 'ring'

Yes. try bypassing the router by plugging your phone directly into the phone jack, and have someone send you a test fax.
What happens it even now days, the majority of fax machines still use analogue phone lines, and the digital signal will mess them up, mostly because the digital lines have less power and current, and do not signal the recieve mode to switch on.
Very common occurance. Glad you mentioned the router or I would have sent you down a long winding path LOL.
You are suffering a low signal issue.
The reason you can call out is that you are manually activating the phone or fax when you remove the receiver. When you receive, it taked the proper signal strength to activate the fax modem.