One thing that I don't get in ASOIAF is the fandom believe Rhaegar would make a great king, but what is exactly his accomplishments beside ToH and played harp?
You’re right in saying that his list of hard accomplishments is rather small. We know of only three tourneys that he entered (though he’s said to have been a consummate jouster, so presumably he did enter others), one of which he lost, one of which took him 12 rounds to win, and all three of which had final matches against members of the Kingsguard, who are known to more or less throw said matches to ensure they don’t injure their future sovereign.
(No one can say he wasn’t in a league of his own when it comes to moping, though. The guy spent extended periods of time alone at haunted Summerhall, the place most of his family died horrifically in a fire, in order to brood and write depressing songs, I mean really.)
However. You have to consider his image.
Until he took off with Lyanna, he was the absolute perfect prince. He was scholarly, which pleased the maesters, but also martial, which pleased the lords. He was handsome beyond measure, he sang songs so affecting they made wolf-blooded Lyanna cry, he wasn’t a womanizer nor did he solicit prostitutes, he was allegedly courting lords to depose his father, he did his duty by his wife and not only had a Targaryen-looking son but also a daughter that could be used for marriage alliances, he did the smart and respectful thing by moving his household to Dragonstone instead of staying in King’s Landing, etc. etc. Even the debacle at Harrenhal didn’t irreparably damage public perception.
Without the benefit of hindsight, well, there was hardly a single thing wrong with him. Maybe a little dreary, but then, Jaehaerys I didn’t exactly sound like he was the life of the party either and he’s remembered as the best king Westeros ever had.