But they were sending it into battle with a flaw that could instantly destroy it. It's just the sheer arrogance and wilful negligence that grates me.
Again, the Empire did not know about the "flaw." Remember that the Death Star's tactical staff had to run an analysis of the Rebels' attack strategy before they had any idea what the Rebs were after. And even then Tarkin refused to believe the analysis. Clearly they had not flagged that vent as a critical weakness. They didn't believe the Death Star had a critical weakness.
Yes, arrogance and overconfidence were always the Empire's (and the Emperor's) great weaknesses. On the other hand, as I've mentioned before, a lot of men died trying to exploit that weakness, and the clear message of the film is that Luke's success was literally superhuman. So there's some justification for their confidence.
I'm not talking about completely defeating them, just gathering the Fleet for a one off in order to just decapitate the leadership on Yavin while they've still got the plans before dispersing again to combat any threats and keeping the Death Star away from any real potential threat. Maybe have it swap its place with one section of the Fleet.
Because, first, the Death Star's job was not to stay away from potential threats. It was meant to be a weapon of intimidation, and it can't serve that purpose if the Empire is clearly afraid to put it into harm's way.
Second, not to beat a dead horse, the Empire didn't believe that the Rebels could seriously threaten the Death Star. On the other hand they knew by then that the Rebels could defeat more ordinary ships. So from that perspective it only made sense to send the Death Star and let "more vulnerable" ships sit it out on guard duty.
And third, it takes much less time to mobilize and move one ship than it does to set a fleet in motion. The Death Star could get there quickly enough, with enough concentrated firepower, to destroy Yavin before the Rebel leadership had time to evacuate. By the time the Empire could have put an equivalent fleet together the Rebels might well have abandoned Yavin - not to mention the havoc they could cause in the systems that the fleet left undefended.
So why couldn't Kenobi have helped a bit with the training at all? I mean, if Vader or the Emperor couldn't find him on Tatooine, training Luke doesn't seem that much of a risk.
The only reason the Empire never found Obi-Wan is because he did everything he possibly could to remain unnoticed. It's nice to think that boy-Luke could have been trusted never to demonstrate his force abilities around
anyone, but would you bet his life, your own life, and the future of the Jedi Order on that hope? I certainly wouldn't.
Also, it's clear from the films that Luke's aunt and uncle knew who Obi-Wan was, and who Luke's father was. They had grown to love Luke, and (particularly his uncle Owen) wanted him to have nothing to do with Kenobi, with Jedi, or with the Empire. They almost certainly would have opposed any attempt by Obi-Wan to meddle in Luke's life, and he couldn't possibly pull it off without their cooperation.
Last, but not least, I wouldn't be surprised if Kenobi had developed a crisis of confidence in his own teaching abilities. After all his first and only pupil had been instrumental in bringing down the Republic and wiping out the Jedi Order. If I were him I'd have a certain reluctance to try teaching another young boy after that.
Even though he had defeated Vader and presented a clear threat? If there's a wasp in the room, I know I want to be able to see the thing.
I'm sure that Vader did want to look for Kenobi - and he almost certainly did so for a while. But every other Jedi in the galaxy was dead, and the longer they went without finding Kenobi or even hearing about him, the less likely it was that he was still alive. It's clear in the film that by the time Vader finally senses Kenobi's presence in the force again, he's almost forgotten about him.