Leliana isn't such a great spymaster, huh.

Leliana has three chances to be hardened and if you miss any one of them, she becomes murder kidnappy Lel. My first PT, I agreed that the traitor spy should die. But some people have also decided just NOT to interfere, and the default is that Lel will kill the spy. So, 2/3 ways for you to fail the "First Check." 2nd check is her saying she should have thrown her forward scouts out into the world like canon fodder. If you agree, this can harden her. Then, if you think the nun that possibly may have tried to kill us at the Chantry should be killed, then Lel will become hardened.

So I told her the spy should die, but told her NOT to make scouts into canon fodder AND told her not to kill the nun. She kills the nun anyway (because I failed the first check) and then proceeds to kidnap someone's son, against my explicit wishes. Well, I don't even think the Quiz can even angriliy disagree, I think they're only allowed to say something polite like, "I'm not certain that's a good thing to do." And then Lel goes into a rant about how it is the right thing to do and the scene ends. 

So, two spy/assassin infiltrations, not telling me Blackwall wasn't who he was, killing a nun I told her not to, and kidnapping a person I told her not to, then I had the audacity to suggest she shouldn't be Divine and she insulted me... I was left with a very angry impression of Leliana and mostly I think she's insane in DAI.

I think you're referring to Ser Temmerly the Ox, who murdered Ser Tamra and who you can "hold for questioning" indefinitely.

When you're meeting with the nobles for the first time, you can instruct Varel to stop the conspiracy in its tracks by taking the nobles' relatives as hostages. It's a very Orlesian thing to do, as Varel says.

So far leliana just feels pointless and boring compared to Cassandra, Cullen and Josephine. If she'd left, I would have not noticed.

The problem is that the way the plots are written they make her seem incompetent because she always seems to find out the truth about someone just after the event.

So she only gets around to checking out Solas' backstory towards the end of battle against Corypheus, thus revealing it didn't check out after he has already left. However, this one is allowable since he did seem to be very helpful, taught the Inquisitor how to close rifts, led us to Skyhold, etc, so in terms of the immediate problem (Corypheus) he wouldn't have featured high on her list of priorities.

Where I was rather more irritated was concerning the elven agents of the Qun that appear in Trespasser. Now she was well aware of what went on in Kirkwall and how many of the elves there sided with the Arishok. She was also knocking around Duke Prosper's hunting lodge during MoA, so should have picked up on the fact that her old acquaintance Tallis was also a Qunari agent.  Gatt is also an example of an elven Qunari agent that we meet.   If we don't side with the Qun and our castle is infiltrated by Qunari assassins, disguised as Inquisition soldiers, this should also have set alarm bells ringing. Yet, despite all these things and the fact she knows these elves came from Kirkwall, she didn't think to check out their credentials, particularly once the immediate danger from Corypheus had passed. Then we discover that in addition to Qunari agents, the Inquisition is also riddled with elves working for Fen'Harel. It made me wonder if there were any genuine elven recruits in the organisation. Since they all up and leave to join Solas after the end of Trespasser, the answer to that would appear to be none.

Looking much further back, the security at the Conclave was frankly awful. There had already been an attempt on the life of Justinia before the mage/Templar war erupted. Yet our elven and dwarven Inquisitors can apparently gain access to the Conclave with no questions asked, not to mention Corypheus and his Warden accomplices. Justinia must also have felt that things might get out of hand considering she had already drawn up the charter to form the Inquisition should a peace settlement not be reached. Why then was Justinia not better guarded? Why was access to the Conclave not restricted and the attendees not carefully checked before allowing admittance? Why weren't there agents throughout the area keeping watch for anything untoward? If the nature of Haven did not allow this, why did not Leliana as chief of security insist that Justinia hold the Conclave in a more secure location? (Haven made no sense to me since it did require the mage rebels to cross the length of Orlais in order to attend, leaving behind their original stronghold) For that matter why did both Leliana and Cassandra need to go in search of Hawke, thus leaving Justinia without both her left and right hands at a crucial time? This is why Leliana appears to be incompetent as spy master.