Ahh, the common mistake of confusing tears with weakness.
I'm not too knowledgeable about Western's philosophy when it comes to leaders, but I can assure you that it's not the same thing for Asian leaders and soldiers / citizens. There were plenty of great kings, generals or strategist described to be crying, bowing down - and in fact, modern politicians adopt this as well (regardless whether their tears upon delivering grave news are faked or not). While some of these cases were obviously exaggerated or even fabricated, they were there because of a simple reason: people liked that. You brought up Churchill and Thatcher not shredding a tear, but then there is Zhuge Liang who cried at battle of Jieting and upon executing enemy's commander - yet he was considered one of the greatest commander in China history (and way better than both Churchill and Thatcher). Western-wise, Eisenhower cried as the paratroopers departed since he knew they had a great chance of dying. If we are to go a bit more modern, Obama apparently cried plenty of times as well - Was he weak? Wasn't he considered one of the greatest US president in the last few decades (and ranked 12th of all time)? Yes, he was, better than some who never shred a single tear. Crying isn't automatically make you a great person, but it doesn't automatically devalue your character either. What important is what makes you cry - and Anduin, according to Blizzard, wasn't crying because "he was a wreak". Shredding tears isn't always the sign of weakness - and definitely not in this case.
It's true that tears don't save life. However, meaningless bravado doesn't either. And it's hardly a sign of being "enslaved to their emotions". That'd be true if Anduin just cried, did nothing else and let his sadness affect his decisions. However, he didn't. He cried tears of compassion upon seeing people - both the Alliance and the Horde - being brutalized and killed. The guy was portrayed as a passionate and gentle soul, after all. Yet, he didn't try to run away, he didn't sound the command to retreat - he focused his mind and power to summon that massive dome of healing to heal people up and gave the order to continue the assault instead. He was emotional, but he knew not to let it affect his decision and do what he believe is right instead. Blizzard called it part of the rite of passage to become a king and to incorporate the spirit of his father and the Alliance into himself. Apparently, that raised the morale of the army, so I'm pretty sure all was good. That's why we should look at the reason behind an action instead of just the action itself.
(On a side note, that dome wasn't a DeM either, he already created that dome of healing before, just on a smaller scale to save Varian from death)