So your saying just because Anakin never said on screen he wanted to give his lightsaber to his son, that makes Obi-Wan a liar? And yes Obi-Wan did take it upon himself to train Anakin, even if it was his dying master's request he still went through it to train him.
Because Anakin and Obi Wan never talk about his kids. Obi Wan doesn't even know they are Anakins kids until Anakin left for Mustafar and then he hides aboard Padmes ship. The next time Anakin and Obi Wan are together they are fighting to the death so... yeah that could have never happened even off screen.
As for Obi Wan. following his masters request is NOT taking it upon himself. Thats doing what he is told. If he took it upon himself to train him he would have been the one suggesting that Anakin is trained.
Also what about "No my father didn't fight in the Wars he was a navigator on a spice freighter" "That's what your Uncle told you he didn't believe in your fathers ideals and thought he shouldn't get involved"
1. Lars never met Obi Wan until when Obi Wan brings Luke to him.
2. The Clone Wars hadn't started so there was no war to go to.
3. The idea that the conversation would have had to been "OH hi I just met you I don't believe in your ideals that your a Jedi Knight you shouldn't go off to war that hasn't started yet instead you should stay here on a moisture farm with me" is just ridiculous
4. Even if they did have that conversation how would Obi Wan know about it?