I think my final thesis with regards to Fire Emblem Three Houses having enough staying power that a character with as nothing as a personality as Byleth won CYL twice is that, inadvertently or not, Three Houses was a game that was tailor-made for fandom. The story is intriguing enough to hook, while sloppy enough for people to discuss and try to fix plot holes. The main conflict is nuanced enough for people to be divided on a "right" side and splitting the story between four routes necessitates engaging with other people's content if you don't want to invest hundreds of hours into the game. The genuinely very good worldbuilding and character writing are the perfect setups for people to expand upon characters and locations in fanworks.
Meanwhile the weakest parts of Three Houses, the map design and gameplay, don't matter as much in the fandom space. Who cares if the monastery is tedious? That's where we hear "We killed Ferdie, Professor, he used to be our friend.", a line that people still bring up to this day. Who cares if the maps are bland and the gameplay is boring? They are simply the means to an end for advancing the story. At the end of the day, these critiques are drowned out by theories and fanfics and discourse.
Naturally, we have to contrast this with Engage. Engage basically excels in everything Three Houses didn't and lacks in everything it did. Engage has, I would argue, some of the best gameplay in the series. It's maps pair well with its mechanics and encourage varying solutions to problems. Its skill system allows you to create strong, capable units, but not anything too broken without heavy investment. This is a smaller point, but Engage also just looks really nice while Three Houses is a notoriously ugly game.
Where Engage falters is in its story and worldbuilding. The story is not bad, per se, but it is very simplistic with extremely obvious theming. We learn very little about the four main countries beyond their ruling structure, primary market, and climate. Like, what does Firene got going on beyond tea, flowers, and being bountiful enough to generally avoid conflict? While the characters do have substantially more depth than most people give them credit for (seriously, you should not be equating them to Fates characters), most of their interactions and backstories are difficult to access. A faster way to grind supports was locked behind a post-launch update, and even then it was way slower than being able to have meals with ten people in Three Houses. The DLC story provides some of the most heart-wrenching lord/retainer dialogue in the series, and a really fascinating look into the motivations and philosophies of the lord characters. The fact that in a barren Firene, the hawkishness you see in some of Celine's supports is realized in her pushing Alfred to invade other countries for resources is makes a lot of sense. Unfortunately the DLC is locked behind a paywall and content that is generally harder than your average FE fan wants to deal with. All the stuff for a fandom to really sink its teeth into either isn't there or is difficult to access, and since discussions on game design aren't as immediately appealing, they kept on talking about Three Houses.
There are a couple of other factors, too. Three Houses was always going to benefit more from the Switch Effect due to it being the first FE game on the system. Engage, I think, was also hurt by the initial negative discourse regarding Alear's design. That being said, I don't think differences in sales figures and a bad first impression don't fully explain why Three Houses is discussed so much. I think it has sort of a perfect engagement cycle. It had a strong enough initial presence that it created a game-specific fandom. This fandom is in turn discussed by the series fandom, which then fuels the Three Houses specific fandom. FEH only adds to this by making Three Houses a consistent staple in seasonal banners and CYL, much to a good chunk of the community's chagrin. Basically, Three Houses gets new discourse every couple of months. Until FEH reaches its EOL, which probably isn't going to happen for a long time, and another, more compelling game comes out, Three Houses will remain a juggernaut in the Fire Emblem community. Truly it is one of gaming's greatest mysteries how sometimes the game that is just worse is the most popular one and you just have to deal with it
