Being too late for the ETD wouldn't prevent GSX from working, it will only prevent it from getting the Simbrief plan, so things like the passengers count or the planned fuel quantity won't be used, but GSX would still work normally without Simbrief, and surely Pushback will always work.
That is, unless that message is not coming from GSX, but from the airplane itself, which is then preventing further operations if you don't have a Simbrief plan.
