Not an expert, but intuition suggests this probably isn’t true.
If an engine is going to fail spontaneously it’s almost certainly going to happen at high thrust, not while at idle or very low thrust values during taxi.
Fair enough. But delaying engine start-up until the aircraft has nearly finished taxiing wouldn’t have any effect on the way the engine operates during start-up. It just means it would spend less time at idle or near-idle speeds during taxi.
Engines experience issues when changing speeds (especially start-up) not when at steady thrust output.