Just adapting the story of the games was never going to work, there's so much more than just the main narrative that any adaptation is always going to feel incomplete.
The P3 movies proved that it would have worked with a competent director + budget. Obviously they're not perfect (minor pacing issues) but it tells the story in a fairly complete way while improving
The first Persona 4 anime suffers from issues with the director (Kishi's forte is comedy, not drama, and it shows) and a low budget. Jun Kumagai wrote the script for some of the better episodes, and when he was given the chance to fly solo under better conditions with the P3 movies, the results were really good.
The Golden episodes seemed almost like a filler afterthought tho. Like they were meant to supplement episodes, but they failed at that because most of them were even worse in quality than the original anime. :\
The episode of P4A where they managed to fit in all the side-social links was really funny. I also thought the episode on Ai/Kou's social links was pretty funny too. Whole show felt pretty genuine to me. Also liked how they did Mitsuo's boss fight.
I just felt like they did a good job of nailing Yu's cool/goofy nature. As the protagonist of a video game, he really just doesn't give two fucks about anything. It's great.
Yeah, the anime had a pretty good writing team, and it's actually fairly faithful to the source material while providing it's own take on the protagonist. It's a shame the production values aren't higher. Seeing actual fight scenes in the P3 movies was a huge improvement over the "disembodied ghost does attack" style "fight" scenes that were present in the P4 Anime and Trinity Soul. Makes me wish we could get an Arena anime with some really awesome fights.