You're the only one here trying to claim she doesn't qualify as a Lord. Creating a debate yourself doesn't really count.

The Fire Emblem fandom is filled with debates about things like this. Just because you haven't seen anything about them on the relatively relaxed and shitpost-heavy mobile game subreddit doesn't mean something hasn't been an ongoing debate among people more knowledgeable than you.

Fire Emblem has a long tradition of various established archetypes that factor into both gameplay and story. While Lucina does have access to the Lord class by default, that does not necessarily have anything to do with whether she fits the archetype. The class happens to share it's name with the archetype, so it's reasonable to be confused, but you don't need to be a dick about it to the guy with the Camus flair who knows what he's talking about.

False dichotomy.

Actually, no. It's not. Whether or not a unit has the Lord class by default is a binary variable. It is either true or false. It is also true that some members of the Lord archetype do not meet that qualification; therefore possession of a "Lord" class is not a good metric for determining if they are a member of the Lord archetype.

She's clearly important to the story as a "retreat" unit. Probably the 3rd most important in the Shepherds after Chrom and Robin.

There's a difference between informed relevance and actual relevance. The game and marketing tells us she matters, in the same way that Mass Effect 3 (if you're familiar with it) tells this one sub-villain Kai Leng matters. He's treated like this super badass cyborg ninja threat, but if you sit down and take a few seconds to think about everything you see him do, he actually sucks harder than just about anybody in the game. Everyone, including your sick, crippled friend kicks his ass given the opportunity, and any small victory he sees is assured by literally an entire army showing up to bail him out. Lucina isn't quite this bad, but the point here is that there's a difference between the game showing you somebody is important, and just swearing to you that they are despite all evidence to the contrary.

The most pivotal thing that Lucina does is tell Robin that "shit gets fucked if you do this." So Robin, after some deliberating, eventually decides to not do that, and being as OP as he/she is (both stat, and lorewise) this difference swings the entire fate of the world. Lucina is a messenger here, a job that could have been done by any of the other kids, like Owain or their potential siblings. Sure, Lucina can have credit for saving her parents, but all the other kids do that job just as well, without their dad being unkillable in progressive gameplay. Everything that changes is a result of Robin's actions.

It doesn't work that way

Actually it does. While Nintendo saw fit to split the same character across both of their appearances, it's still the same character. Every vote that either Ike got is a vote for Ike. You could maybe make a case for Selena, Laslow, and Odin being different from Severa, Inigo and Owain, but Ike is definitely the same character. He does not have the Young Ike, Toon Ike, Regular Ike split that Link does.

Creating a debate yourself doesn't really count.

Let's revisit this point quick.