Not familiar with the Airbus logic - Boeing handles it a bit differently. If there is a detected engine fault after start, there is an EICAS "L/R ENG CONTROL" message - which is a no-dispatch condition. However, that message is inhibited above 80 knots - the logic being that at that point there is little that the pilots can do about it - and while the fault may be 'non-dispatchable' - it may not cause a shutdown so putting up a message could result in a nuisance IFSD. The EICAS message will appear after landing, and must be cleared prior to dispatch.
That being said, there should never be a case where an EICAS message related to engine health gets set without an associated maintenance fault also being set so that maintenance knows what to fix prior to return to service.
Unfortunately, at EIS of the 747-400 (~1988-90), we did a real good job of setting these messages nuisance, in effect training the maintenance people that if they cycled power and the message didn't come back, no further action was required. While the nuisance message issue has been pretty much eliminated, old habits persist...