Taylor rattled off two options for if she has to do things without the order. One is watching Harry's back for anyone interested in planting a knife there the other is going hunting. Naturally dumbledore hates both of these options but lacks any morally acceptable means of forcing Taylor to pick a third choice. He can set her to work with the order but then she'll turn things into combat missions or decide he isn't ever going to step up at which point she goes rogue anyway.
If I were Dumbledore here I would (a) set her to work with the order, (b) team her with people she might want to and be able to convince to go her way rather than Dumbledore's, (c) constantly send her on missions investigating and/or capturing people suspected to be
imperiused or blackmailed or otherwise non-willingly working with the Death Eaters.
It's a delaying action, but (1) these targets are a real concern, (2) her super-surveillance skills may work well for figuring out what's actually going on, and (3) it will force her to play by the book if these people might be varying degrees of innocent (and she doesn't want to look bad in front of her teammates). (4) She would be a reasonable choice for someone to put in charge of discovering new, possibly-compelled Voldemort followers, and (5) he might actually believe that seeing enough actually-somewhat-innocent forcibly-recruited people would convince her that holding a trial to prove things beyond a doubt is worthwhile even in the case of his longstanding supporters.
Snape claims credit for the death of an order member. He wouldn't risk that if it wasn't partly true because word would spread among the death eaters and if it was an outright lie…. Anyway.
I think it's very plausible that e.g. Snape and another Order member showed up on an op, ran into a Death Eater ambush he doubted they could escape, and he stunned his partner in the back and said "hey, I caught you an order member!"
Or he came across some death eaters searching e.g. "Charity Burbage" in muggle telephone books and said "oh, I know where she lives, I can take you there." (Because he didn't think he could save her without exposing himself as the one who tipped her off, and so might as well make use of her death for rep.)
Sets a teenager with no training an important espionage mission instead of letting Moody force the memory from slughorn.
To be fair, Moody is, even if retired, an Auror, and does not have Harry's shield of fame, and has a much more confrontational attitude. If Moody doesn't kill Slughorn, it's very possible that Slughorn hits him/the Aurors with enough backlash that it would be a net negative.
Yes, Harry was still likely not the best choice. But I think Dumbledore's real failing here might be not giving Harry any real advice or training. Even just saying 'oh, you could bring Pavarti with you, she's good at ferreting things out of people' or something. (That may be terrible advice, idk.)
But I think Dumbledore kind of wound up treating Harry as though he weren't an untrained teenager and was magically, fated-ly competent at whatever anti-Voldemort activities Dumbledore thought should happen--and to be fair to Dumbledore, Harry's track record is really not evidence against that.
Hands out critical information one drop at a time over months.
I'm not sure how much of that he was really sitting on to slowly dole out. For example, I think he found the seaside cave with the Locket right before going, and I think that implies some of the other stuff was pretty recently validated.
And then because dumbles changed the subject but made Harry feel better, Harry just drops that completely separate concern and goes away happy. I remember that bullshit because it was one of the stand out scenes on my "want to strangle the author" list.
...This seems very realistic to me? Teenager is angry, has a legitimate concern. You pacify them and compliment them and calm them down and give them something else to think about, and they completely forget about the legitimate concern that originally made them angry. Core technique of parenting?