'Coffin corner'??? What has that to do with this accident? Now, depending on your meaning of 'Service Ceiling' (eg 100fpm residual rate of climb, max alt for oxygen considerations or max altitude according to WAT limits), I don't know if they bust 1. They did not bust 2 but they did bust 3.
So, that expression is meaningless. Surely any pilot, knowing that he has climbed above his WAT limit would lower the nose, wouldn't you? That is what we mean by 'Max Altitude' and it is probably labelled something similar in the FMC.