Anything raw metal, exposed to the user or sometimes not, should be connected to the incomming frame ground, via the center pin of the power cord going into the cabinet, aka earth ground.
This includes the exposed metal on things like the: Coin door, control panel, speaker/vent grilles, power switch metal plate, fluorescent lamp frame, isolation transformer housing, monitor frame, back door kill switch support plate, etc..
Anything not exposed to the user and part of a signal loop back to the gameboard should have their grounds shared and connected across all ground points on the board's harness.
This includes everything like: Joystick and button leafs/microswitches, Trackball/spinner optical boards, service switches, coin switches, etc..
As a generality, the video cable (output) signal grounds normally are separate from the 'input' grounds, and you can safely just send the 1 thin trace to the monitor, because the amount of current is so small driving the video, the return lead doesn't need to be great. In some cases it might be that that ground is separate from the other signal grounds, in other cases, the grounds are crossed anyway so you can connect ground (for video) to any ground on the connector.
However -- Given a separate video ground, I would never puposely cross a video ground to the main input grounds. You could end up with interference if they are designed to be isolated. And more importantly -- NEVER run the main input grounds into a single video ground trace as you will definitely end up with too much current into a single small ground connection and fry the connector or board.