What you seem to forget when you say that Robert should have looked towards a more diplomatic solution is that Robert wasn't given even the beginnings of a choice. The Mad King called Robert and Eddard to his court, and given how the Mad King had killed Rickard Stark, roasting him alive, while his son strangled himself in desperation, well, it was extremely unlikely that the two youths would have gotten anything but a beheading. Remember that Jon Arryn then decided, on his own, to rebel, and it was a question of survival back then that Houses Baratheon and Stark join with House Arryn. I find it hard to believe you accuse Robert of going too far when he actually couldn't do anything else at that point. There was no 'diplomatic solution' that didn't involve death where the Mad King was involved by that point.
Aemon had taken the Black. He couldn't be King anymore, no matter what.
As for Rhaegar, tell me: didn't his taking Lyanna (by force of willingly, doesn't matter here) and disappearing with her cause Brandon Stark to go to the Red Keep, make a scene, which the Mad King followed with a series of gruesome killings and insane demands, which caused three major houses to rise in rebellion? It did all of this.
Rhaegar, took a woman away without saying what he was going to do with her, cheated on his own wife, recklessly started a situation which led to a continent-spanning war... and we're supposed to 'not judge him too quickly?'
Well... no. I think he deserved to get killed. Smashed down by Robert's warhammer. He doesn't deserve even the tiniest inkling of respect. He deserves only loathing.
That's how I judge him. Whether you agree or disagree changes nothing to that.
But his dad was insane, and it did start a war. The guy had no business just whisking a Stark who was bethrothed to a Baratheon without even saying anything to the Starks and Baratheons. Actually, that's the definition of a kidnapping. What DID Rhaegar think it would do? Did he think pissing off two Major Houses was smart? The man was an idiot at the very least. His idiocy directly led to a war.
He deserved to go.
Not everybody liked Rhaegar. Some, including Stannis and Eddard, found that he was reckless, irreponsible, and dishonorable. Given that Rhaegar, for good or ill reasons, kidnapped a bethrothed woman while MARRIED to another one, he was reckless, irresponsible, and dishonorable. Its certainly exaggerated to say 'everybody thought he was awesome except Robert'. I just reread the first book, and he was seen as a selfish man who was in the wrong.
As for Eddard and Robert uniting the other Houses? Those guys were 19 back then. They could call on their own bannermen and rallied their own House, but the Tyrells had plans of their own, the Greyjoys just wanted the other Houses to weaken each other, the Martels had filial reasons to remain loyal, and the Lannisters were... being Lannisters.
Your opinion isn't changing. Neither is mine.
Aemon was Aegon V's older brother, a man who ruled well for over 25 years. The Mad King is actually his grandnephew. By the time he truly was mad, Aemon was around 80.