Well, remember, Jaime wasn't executed or banished or really punished in any way.

He was still a Kingsguard to Ned's best friend after killing the previous King, in by far the least noble of ways in their eyes, was an enormously pompous asshole from a family that was untouchable because of their wealth, and there were the obvious rumors that he was banging Cersei that pretty much everybody knew about.

Ned hated every fiber of Jamie's being, which led fed into his own faults as a human, and eventually led to his death.

Ned wanted Jaime to admit he killed the King just so he wouldn't die. But until Jaime did that, which he never would, all Ned could do was be an ultra-condescending asshole to a guy who was a huge, untouchable, prick.

I think there was also a belief that he did it at Tywin's request. And nobody was going to fuck with Tywin, so they kept their mouth's shut.

Brienne was the first person he told and in reality he was pretty vague about it all. He gave some possible facts, but didn't really go far into what he was actually feeling.

But, honestly, at his core, Jaime is an ultra narcissist who knows he killed Aerys out of self-preservation and leaving out some of the true bits kept people talking about him and paying attention to him. Jamie didn't care about glory. He enjoyed fighting and killing, and if he had to be known as the Kingslayer to keep doing those things, so be it. Which I think Ned subconsciously knew and it's why he always looked down on him more than just calling him a, derogatory name.

I don't think most people knew what Aerys was trying to do when Robert's army got there. Even Robert and Ned didn't know the truth.

To them, Jaime did in fact stab his king in the back to end the war and that was it. His honor was already so far gone that he didn't really tell anybody why he had done it. Also, his ego and the believability of a king telling them to "burn them all" probably stopped him from saying it.

The only reason he wasn't executed after Robert took power was because of his last name and who he shared it with (Tywin).

Also, Ned was loyal to an absolute fault. He didn't care for any real reasons. Jaime broke an oath and that was enough to despise him for (which was Ned's biggest fault).

They made a documentary about the area like 5-ish years ago. Atomic Homefront is what it's called IIRC.