There are moments/episodes that each of the captains are annoying, or when they lose my sympathy.
For Picard, it's when he gets really self-righteous, and has that 'I hate you. I'm disgusted by you' expression.
Picard had too many self righteous moments.
The other captains had their sanctimonious moments as well. But Picard was the one who came across as a patrician, imo. I thought he was a stiff and he seemed unapproachable. His unease with kids was an example of this. He wasn't my favorite captain.
But when Picard was completely out of his element, like when he was wrestling in the mud with his brother, his actions could be really amusing. And during those rare occasions, as with that scene, he can be a sympathetic figure.
Also, not even Picard was perfect on a moral level. There are instances in which he failed, morally. Homeward would be an example in my book.
I would add "Suddenly Human" to that. I would say that Picard failed in that instance too.
The episode followed the typical TNG plot: conflict ... Picard giving his holier than thou speech at the end ... conflict solved.
Not only was Picard wrong in returning the kid, Jono, to the Talarians. The speech that he gave was just outrageous.
Picard basically blamed the victims for crimes committed against them. It wasn't just his excusing Jono for stabbing him. Picard accused himself (and Starfleet) of committing a crime by trying to persuade Jono to stay and be reunited with his biological grandmother.
Picard essentially excused the Talarians of the real crimes that they committed. The Talarians killed Jono's parents and then took Jono as war booty. They kidnapped the kid. Picard twisted the situation around. It was outrageous.
Interestingly, DS9 also had an episode, "Cardassians", that dealt with child custody.
The two situations weren't exactly the same, but there were similarities. In both cases, the kid bonded with and wanted to stay with his adoptive alien parent. Rugal's adoptive father took Rugal in, in good faith. He adopted a kid whom he thought was abandoned, which was opposite of the Talarians who outright stole the kid.
Ultimately, it was shown that both kids were stolen (it was Dukat's shenanigans that got Rugal "orphaned") from their biological parent, yet Picard and Sisko came to different resolutions.
The two episodes aired about 3 years apart. It was a shame that one, the spinoff, got it right but the other, TNG, got it wrong, imho.