Depending on what you like about Legend of Galactic Heroes I have a few suggestions for some other anime you might enjoy.
Rose of Versailles: This is a historical anime set in France around the time of the revolution. It has a definite sense of place and quite a lot of political intrigue. It also treats questions of ideology, related to how the main character Oscar, a noblewoman raised as a man, reacts to the revolution as it intersects with her personal relationship with the royal family. It is also a very influential shoujo manga which inspired many series to come after, but I think it is absolutely worth a watch on its own terms. While the galactic empire is mostly Prussian-inspired there is a slightly similar feel between the historical France as presented in this show and the galactic empire as presented in LoGH.
Saiunkoku Monogatari: This has less ideology (as far as I remember), but has a lot of politics and palace intrigue. It is set in a Chinese-inspired palace court and follows a young woman, Shurei, as she becomes a palace bureaucrat, a position which women are traditionally forbidden from. It does have a bit of a reverse harem setup, but the focus is mostly on the court bureaucracy and how Shurei handles various economical and bureaucratic issues.
Legend of Arslan: I have not seen this show, nor read the books, but it’s a fantasy series based on books by the same author as LoGH, so you might enjoy it. Will look into it myself at some point.
Twelve Kindoms: This is another series of fantasy books which has an anime based on them. They follow a bunch of different characters, but the main focus of the anime is Yoko, a girl from modern-day Japan who is taken to this world. The fantasy world is quite weird, but with heavy Chinese inspiration which is a nice break from the traditional western inspirations. However, the main reason I am suggesting it is because it also contains a lot meditations on what it takes to be a good leader and how to rule. If you find Yoko annoying initially, I promise that she has a lot of interesting character development! On a side note, I would be interested in some suggestions for similar stories, particularly setting-wise.
Crest/Banner of the Stars: If you like the military aspect of LoGH, this show does a very good job with tactical space battles as well. I think the Abh are rather interesting as a society, but if you expect the nuance of LoGH you will be somewhat disappointed. Also the show focuses a lot on the main characters and their relationship which means you see stuff though their point of view giving a somewhat biased view of the conflict. I think the books are a lot better about this, but watching the anime you might get annoyed and feel like you are made to cheer for the Abh, despite them not really being good guys, because their enemies the United Mankind is presented as almost cartoonishly racist. I still think it’s a good watch though.
Berserk: The manga or the 1997 anime. The part adapted in the anime specifically has a lot of political intrigue and some discussions related to leadership, war, the relationship between ruler and ruled as well as political violence and its justifcation (or lack thereof). Content warning though, it is much more edgy than LoGH and contains sexual violence.
Irresponsible Captain Tylor: This is mostly because you mentioned LoGH. This show is a comedy which parodies a lot of the elements of military sci-fi, space opera etc. and the main character Tylor feels like a pretty deliberate, but loving parody of Yang Wenli. Despite the comedic nature of the show, it does have some underlying themes related to war, pacifism and how seriously one should take life. The main premise is that Tylor is irresponsible as implied by the title, but somehow always comes out on top either because of extreme luck or because he is actually a genius. It is never made entirely clear which one is the truth.
Kemono no Souja Erin and Seirei no Moribito: These are perhaps slightly different, but have a definite sense of place. These are both based on novels by Uehashi Nahoko. Seirei no Moribito only covers the first novel, the later novels have much more political intrigue and are very good, but unfortunately they have not been translated to English as far as I am aware. What I like about her writing and these anime shows is that she is very good a creating worlds that feel real and lived in, with folklore that feels like stuff actual people would believe in, but which just so happens to be connected to real elements in the setting (It probably helps that she is also a professor of Ethnology). Both series deal with the intersection of politics and myth and how royal families and bureaucracies utilizes such myth. Erin covers two books out of four books, but these two are a finished story, with the latter two written later. In addition to the other topics mentioned above it is concerned with the relationship between humans and animals and I think it does a good job of portraying the main character Erin as someone who is ideologically principled, but has to deal with situations that challenge her morality. Fair warning, the Erin anime does contain a bit of childish humor every now and then. Again, I would be interested in recommendations for books or shows somewhat similar to Uehashi Nahoko’s writing.
Monster: Not super similar, but it wrestles a lot with moral questions in a way I feel would appeal to those who like LoGH. It is also rather meticulously plotted and does a good job utilizing the German and central European setting to create a sense of atmosphere and place. It is also my second favorite anime after Legend of Galactic Heroes.
Shin Sekai Yori: This one is also somewhat of a left-field suggestion, however it is another examination of how to structure a society. In this case how to deal with every single member of that society essentially being a walking nuclear psychic bomb. I really enjoy the unsettling atmosphere of the far-future agrarian society as presented. It is relatively dystopic, but also makes sense from the point of view of those who set it up. However, it is also revealed to have an even darker side and is a very good example of a show with sympathetic protagonists who are ultimately…probably morally wrong, although the antagonist is also quite grey. Like many of the others I recommended, it is based on a novel.
I was wondering, since you seem to know a lot about Japanese media if you have any opinions on 帝都物語 in relation to this type of millennial Japan Occult media. I’ve been thinking about reading this series, but haven’t quite gotten around to it yet. As far as I can tell it is quite influential? I guess I am also wondering whether it is actually any good. The easiest way of course is to just read it…but would be interested in what others have to say first.