The life of one man, when weighed against the possibility of the Inquisition failing, isn't worth it. Any chance that he might falter when it matters is too great a risk to take when the fate of the world is in the balance. In the grand scheme of things, it doesn't matter a whit what happens to Cullen. All that matters is closing the Breach and killing Corypheus.
Thing is, you've been given absolutely no indication that he CAN'T do the job, that's my point. Experts are telling you that he's fine. You don't suddenly upend your entire command structure because he's having a bad day, which is what it is. Now if there'd been evidence that he was slipping, or Cassandra had told the IQ yes, something is wrong, you might have something, but it's not the case. It's a solution without a problem.
When you're in a position of power as great as that people are disposable tools, it naïve to think otherwise. The Inquisitor is no different; all the soldiers the Inquisition commands are disposable tools. How many hundreds or thousands are thrown at the walls of Adamant fortress just so we can break in? The simple fact is that human (and elven, dwarven, qunari, etc. I suppose) lives are the currency great powers gamble in their games of power. So Leliana kills a few dissenters to stamp out rebellion and assert her power, if it leads to fewer deaths in the long run, it's a good gamble.
No it's not, because if you think that way you're going to make the mistake of seeing killing as the first option.
The soldiers are sent to Adamant because there's no other option, however, they aren't just "thrown" at the walls. Sacrifice is needed but a good commander doesn't throw lives away needlessly nor does he reach for a sledgehammer when a smaller one will do. Good commanders don't just throw people at a battle. Good commanders are sneaky, good commanders do everything they can so they don't have to sacrifice so many lives. The trebuchets, sappers to undermine walls... all things done so the Inquisition doesn't just throw people at the walls.
We get an even better example of it in the montage of the Arbor Wilds. Harrying the enemy forces, hit and run strikes, whittling them down til the main force gets there. Tactics like that save lives in the long run.
No, there's no guarantee that a soldier isn't going to die when he goes into battle, but that soldier's commander OWES it to him not to get him killed needlessly or for some worthless objective. Someone who doesn't see his soldiers as people, with lives and families is more likely to do that.
Take a look at the really good generals throughout history. Patton, Rommel come to mind. They generally aren't the type to see their people as disposable parts.
As for Leliana, the traitor conversation? She's making a decision without any time to consider her options. Her first automatic response was "kill him" rather than considering how she could turn this around. In fact, I find the whole situation implies that she's doing it more out of anger at Butler's betrayal than any cold rational choice. She had no idea why Butler killed him, who Butler was working for... she would have thrown all that away.
Then there's post Haven. Unless she had watchers so far out that they saw the army coming about the time the IQ went up to seal the Breach, maybe she'd had something. But the implication was that they weren't that far from Haven, in which case a few minutes against a force like that isn't going to mean jack. The Inquisition wasn't going to beat that force. And an hour warning wouldn't meant much. In fact, it might have made things worse depending on how they were retreating, if Rodrick said anything, and if anyone was willing to listen. It wouldn't have really mattered in the long run.
With Natalie, her patron was as much of a threat as that idiot that Sera sends you to. A posturing windbag. To be watched, certainly. Have to make sure things don't fester. But killing Natalie gains you nothing. She's been neutralized as a threat. You now know that the Grand Cleric is taking actions, stupid ones maybe, but actions, and it's not as if Natalie knows anything important about what Leliana and the IQ are doing. Killing her at that point is just blood for blood's sake. Especially considered how much of an incompetent schemer she is. And while there's an argument for taking stupid people out of the world, Leliana would never get any sleep.