I recently purchased F-15 City War (American Video Entertainment). The cart is in good condition, and I've cleaned it, but the system will not play the game. The system I have is a NES-001 with a disabled lockout chip (Pin 4 has been cut.) Other games run fine.
Bad ROM or write to game? Not sure, but that sounds like the only thing that could actually cause it.
Replace the caps and check for cold joints and If that chip to the right is indeed an eprom it might need to be rewritten(but this is if all else fails).
I have a Metal Fighter that hasn't worked for some years. When I opened it, it had 1 EPROM in the PRG side so I guess it can go bad after less than 10 years. I still need to remove it, reburn a fresh chip, and solder in a socket then plug it in. Considering how crappy the game was, I'll probably just save the EPROM for something else.
Check the vias as well. I once fixed a Grind Stormer for a friend, where some of the vias had broken. I had to remove the ROM to get access to all of them though.