Fios cable boxes never 'power off', they just switch to a splash screen which floats around and says 'Press Menu To Turn On', so there is always an active HDMI signal on Fios boxes and a maintained DHCP and EDID signal in place.
My guess, is that some sound setting may be getting transmitted to the Yamaha AVR at startup. I used to do this as sometimes customers would screw with audio settings and I always pushed it back to 'surround sound' and 'auto' for the input type. Some sources were analog or some were digital, and depending on selections, I would force that as the input type.
I'm currently running a couple of different Yamaha receivers in my setup, and both work perfectly without any audio problems that I've run into so far. All I do is power them on and off though, I don't switch inputs on them as that is managed externally.
I would think that upon power up, the C4 system should have a proper delay built into the system that would keep it from trying to switch inputs for several seconds after startup. But, typically products just ignore commands if they are in a transition state, it wouldn't keep them from operating properly.
I would reach out to the installation company and let them know you are having some issues, and be specific about what you're running into. You want to be able to recreate the situation on a somewhat consistent basis so they aren't just shooting in the dark if at all possible. There is nothing worse than a rarely intermittent problem. But, I would lean towards some command being sent to the AVR from the C4 system which is switching audio inputs at startup. It toggles through something, then ends up on a dead audio input at some point. I would bet if you used the Yamaha AVR remote and switched your audio input type back to 'digital', it may fix the issue.
You may want to look at the front of the Yamaha AVR and see what information it is giving you about audio when you DO have audio, and then again when you do NOT have audio, to see what is different.
If you have no audio at startup, does switching to another source through the remote correct the problem? That is, if you press 'CABLE' and there is no audio, then you press 'ROKU' (or whatever) do you then have audio? How about if you then switch back to 'CABLE'? Do NOT power off in between. Just leave the system on and switch sources.
I don't know enough about C4 programming to really know for sure what is causing this, and despite having a couple of Yamaha AVRs, it is not a problem I've run into when using my control system. But, I am not sending extraneous commands, and I have logic built in that prevents commands from being sent until after the unit is powered on.