I love this game to death and it is one of my favorite FEs. However, it also has so many issues that kind of frustrate me, but I think it really speaks to the game's quality that it's still talked about and discussed even years later.
Crimson Flower feels unfinished which is a shame because Edelgard is one of the most interesting characters in the game. It has no CG cutscenes, it's way shorter than the other routes, and the plot is really straight forward. According to the devs, Crimson Flower was developed last and wasn't even originally planned to be in the game, which is not surprising at all.
Azure Moon is one of the best FE stories ever to me. It focuses on doing what FE does best -- it's a big character study on Dimitri and an analysis of monarchies like the Holy Kingdom of Faerghus. Dimitri and the setting of Faerghus is deeply, incredibly flawed, is filled with hypocrisy and messy characters, but it's portrayed in a light where it actually feels intentional, and it's all so interesting to think about and watch unfold.
Verdant Wind is okay, but it really doesn't reach the heights of the other routes to me. It suffers a lot from sharing the same plot as Silver Snow, which makes certain scenes feel incredibly out of place. Claude is also probably one of the most hypocritically written protagonists I've seen in a video game. My issue with Claude's flaws is that they all feel unintentional, and Verdant Wind's ending is kind of terrible and it is obvious the entire continent will collapse into chaos the minute Claude is out of the picture, which will be obvious to anyone familiar with conquest and empires throughout history.
Silver Snow is just a lamer Verdant Wind.
The DLC, Ashen Wolves, is really strong as well, and it has a really charming, unconventional, likeable protagonist in Yuri. The final boss, in my opinion, is one of the best in FE as well.
Story aside, I think the gameplay is really good in Maddening mode and it really forces you to think hard about how to build your units, and you absolutely HAVE to use every tool at your disposal and have an intimate knowledge of the game's mechanics. Hard mode is way too easy, however, and New Game + makes any difficulty, including maddening, absolutely trivial. Repeat playthroughs can also be incredibly boring, especially since all routes share the first half of the game with the school setting.
All that said, I do think FE3H has a lot going for it, with some of the best characters and music in the series, and it has all sorts of QoL changes that make it a lot of fun. I would recommend it to anyone who's a fan of JRPGs or games with interesting stories.