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If the remote is 5 years old, the batteries in it are fresh, the metal terminals in the battery compartment are cleaned of all possible corrosion until they are shiny-bright, but the remote won't start the engine until the key has been used (...I don't quite get the picture - when the key is used first, then the engine is shut Off, then the remote is used soon after, perhaps?...) then it is possible that the remote is 'broken' - slightly, if it still works at all in any situation - with sheer age.
When the remote does work, is it with the key still in the ignition and turned, or with the key removed from the ignition? it may be simply that the remote has been deauthorized, if it still doesn't work when the key is removed from the ignition and the doors are locked. You may have to run the reconfiguration (that you did, or that the dealership did perhaps) when this was a new system. That 'reconfiguration' would cause the alarm/remote-start module to once more recognize the remote as authorized to start, open, etc. the vehicle without the key being used for access or in the ignition etc.