Tachi is also a (relatively) normal outsider providing an additional and more broadly relatable perspective on everything - notice she never spends any time thinking about herself as a pervert or anything like that, she's always got a positive view of herself and for the most part other people, and if she has any romantic interest in Kaoru it's pretty minor, never more than a small crush. Her open minded and positive reaction to all the stuff that ends up happening helps the reader accept it, and frames what could otherwise be a disturbing and uncomfortable subject for many people as just a somewhat unusual (and of course titillating and erotic) focus for the shared interests and experiences that build Kaoru and Nana's relationship.
And frankly, Nana and Kaoru really need a third party to kick their arses, and it needs to be someone who knows about their relationship and accepts their interests. The story has already set up the idea that if people knew about their "breathers" then it'd be a disaster - that seriously raises the stakes of having random outsiders giving them the arse kicking they need. So Tachi is the accepting insider who already knows what's going on and can intervene if it's necessary, and help progress the story in ways that would be impractical without her.