No, on a breadbin (with the older bigger PCBs) the +5V supply for the VIC-II (and the SID) is obtained by rectification and regulation (with a 7805) of the 9V AC supply, I don't remember what happens inside a C64C (with the newer shorter PCBs), but might be similar.
shows +5v and +12V. The +12V uses a voltage doubler from the 9VAC to feed the +12V regulator. It states the +12v is used for the VIC and SID chips. I think you would want both +12v and +5V to power it. With not video, what would you be looking at?