Receive same GPIO input signal on multiple raspberry's from one source
Just connect the activation signal to all 7 GPIO pins. 

You can only overload an output. (Where overloading means: taken more current then they can provide). 

Inputs can not be overloaded in hat sense. You can 'overload' them if you put too much voltage on them. But if you keep between 0 and 3V3 the Raspberry-Pi will have no problems.

The signal going into them must be able to drive all inputs but as we have CMOS inputs and, I assume, the signal is very slow, 7 pins is not a big issue.