Yes the BAe146 was built in the snow. Where I fly I am more likely to see ISA + 25 than ISA. But with low freezing level, lower density altitudes mean more air flow reducing the chance of the roll back. Originally temps of -40 degrees C no icing was thought to exist. Now all temps below 0 degrees C Icing are thought to form. Part of the reduced risk for roll back is better air sharing between # 1&2, and # 3&4. My Co now has better equipement to achieve this. Also new fans give us 5 degrees better temp margins. The # 4&5 bearings have been combined to reduce vibration. Not sure this has worked as it should.
As for better engines, my understanding is that BAe do not gaureentee it will solve the problem. The new attachment point on the airframe are very expensive. Not that I understand why the great expense, this is a major factor why the engines are unlikely to be changed.
All acft have limitation. I just try not to fight these ones.