Why did they freeze to death? Arthas survived. Some of his men survived, as we can see with Falric and Marwyn. So why did some of them freeze to death? Arthas got the sword, killed Mal'Ganis, and then made his way back to Lordaeron. How did he do that without ships? He had a way to go back. Clearly he took some men with him. Did he leave some men behind? I am sure had he not been cursed by the blade, he would've attempted to get them all back, but maybe, after he took up Frostmourne, he already didn't care enough about his people anymore.
But why did they just freeze to death in Northrend? I mean Northrend is harsh, but people survive there. People are living there. They have houses. We've survived there for an entire expansion and beyond.
And the burning ships thing is a strategy that has been used in real history by real leaders in order to make sure your troops can't go back and have to give everything going forward. It works, and it has worked in Arthas' case. These soldiers swore to give their life protecting their own people, and Arthas made sure that they would.
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It's not about "having done something bad". They were literal murderers. They would have killed Arthas and his men had someone paid them to do it. I'm sure Arthas didn't have a hard time making that decision, and most people wouldn't have. Especially when the survival of your own people is on the line. Would you sacrifice some hitmen for hire, in order to protect your women and your children at home? If not, then you're a bad leader.