There is a familiar trope that is pretty much everyone has universally agreed to be foolish except for the people who write it, that being "11 year old politics."
Let's make that work in a serious, reasonable story.
Step 1: Harry needs to have a reason to give a shit. When he first arrives at Hogwarts, canon Harry doesn't care about the mundane details of the world he's in. He might find the idea of the Ministry of Magic fascinating in a, "Let's go on a trip to the capitol and explore the big buildings for a day," but the truth is that he's way too caught up in the wonder of magic to give a rat's ass about the specifics of laws and politics. So how do we change that? By tying the two together. We give Harry an interest from before Hogwarts that combines the two. Maybe he found a copy of an old Dungeons and Dragons manual when he was 7 and, in the absence of people to play with, re-read it over and over again, eventually getting past the basics of magic and mechanics and became a superhero about world-building and social systems because that was his escape from the Dursleys and he didn't have actual friends to play with, so he'd just make increasingly complicated worlds. This would give Harry a reason to care about the minutiae of how exactly the magical world works, because it would become something like his escapist fantasies. . Step two: We've gotta make other people give a shit. We've all read the absurd scenes where fully grown adults are bending over backwards to cater to the eleven year old lord potter-blacl-peverell-pendragon and we all know they're crazy, so it can't be that. Instead, consider Lockhart. He was very interested in Harry’s political implications as the Boy Who Lived. Not in legal politics, but in the court if public opinion. Add a few more characters like that who want to exploit Harry’s fame and are willing to give him some compromises to do it, some adult characters on Harry's side who see what the predatory others are doing and will step in to defend Harry from exploitation, and one or two people that see all this happening and think to themselves, "we should probably start teaching him about this stuff sooner than later because he'll be dealing with it for his whole life and we don't want him to be unprepared," and you got a fully rational explanation for why adult characters are treating Harry in a serious political manner. Not because he is a political mover, but because he is a political prize.
Step three: Get other young characters involved a little bit at a time, and only where appropriate. Hermione would be delighted to get involved with Harry's study of politics as way to learn more about the culture she's in, Percy would gladly help out Harry with fending off the Lockharts of the world for a bit of experience and exposure to the caliber of political person for whom trying to exploit a national hero is even an option. Ron has no reason at all to get into politics, except perhaps at the end of second year when he knows for a fact that Lucius Malfoy almost killed his sister and walked off scot free. Etc. . Step four: Start small so you can grow big. Don't start off with Harry on the Wizengamot, start off with Harry trying to juggle the attention of a single Lockhart-esque antagonist with one or two well meaning but not very politically savvy adult mentors/helpers, the grow until he's interacting with the Daily Prophet, then the Ministry, and finally save the Wizengamot for last so that Harry will have a slow political escalation behind him up to that point and then drop the bombshell that a lot of the children of the Wizemgamot members are Harry's classmates, so now the other students at Hogwarts get pulled into the political game only when Harry does the pulling. This evades the trap of all the first years being political pawns more interested in forging alliances than learning magic, but it still gives the children at Hogwarts real political weight because they, much like Harry was, are in the position of being a political prize that Harry can pursue to try and get an in with their parents. Basically, make Harry the Daphne Greengrass of the story, actively pursuing political interests even when others might not be. . Step five: Revel in the realism. No one is going to take Harry seriously. Accept that, and also accept that it means that Harry has the opportunity to take advantage of their underestimating him and thinking they can control him, after he has narratively earned the political skill to do so. Make his age more than just an obstacle, make it a feature.