We'll have to agree to disagree on the first point. In the same of historical accuracy, though, I'll point out that there have been times when Batman was the one trying to kill Joker and Gordon kept him from doing so - threatened him to put a bullet through him if he didn't step down, in fact.
Feeling regretful I can understand; trying to destroy themselves from guilt...just, no. Obviously Dick has a very strict moral code.
it was clear that he would never have taken them on unless he saw the required aptitude for the mantle That depends on what you read. There have been times when Bruce thought that 'he always knew Dick had a gift. Jason... Jason had only rage.' There have been times when it's been implied, if not outright said, that Bruce didn't train Jason well enough.
He didn't even tell her, his so-called Robin, his true name! I'm aware that Batgirl and Robin are two different roles entirely, but he never told Cass his name either.
Batman never neglected Steph's training as Robin. He was training her physically and as a detective. She was out on the streets before she reached the level Dick and Tim and possibly Jason had when they started, but in her case it makes sense in that she was already used to being a vigilante. She had years of experience as a crime-fighter by that point, and he didn't let her out on her own either; she was always either with him or with Cass.
he took on Stephanie as Robin purely to lure Tim back No, he doesn't answer Alfred's accusation. And given Bruce's inability to communicate, I don't think we have to take his silence as a simple confirmation. He didn't answer the Batfamily's questions when they asked him if he'd murdered Vesper Fairchild, either, remember?
Whether consciously or not, he set her up to fail from the start, giving her just enough training to make her dangerously overconfdent in the process. That's a big assumption. My assumption is that he was as usual playing Merry Mixed Messages - "I don't mind your talking." "Stephanie go home." "Excellent initiative." "You disobeyed, you're fired" - the way he always has with her and with every other member of the Batfamily. I personally don't doubt that he'd have taken her back a few weeks down the line, but it's never been Robin's thing to wait and do nothing. Whether Batman is aware of it or not, Robin is meant to disobey him anyway when Batman is going off-tracks. And that's what Steph did when she started implementing the War Game.
Steph's death was a pile of "badly-handled", and as far as I'm concerned, the more they sweep it under the rug, the better. The derailment of Leslie Thompkins' character alone was... ugh.
It seems that he and his cohorts would take on the guilt for the death of an enemy more readily than the deaths of his own charges. The case with Jason's suit has been standing in the Cave since before Tim became Robin. Every single time DC can shove it down the readers' throat that Batman is Angsting and Feeling Responsible for his charges, they do - often with a helpful shot of aforementioned Case.