Let's look at other female characters people suggest:
Lyn: One-shot Fire Emblem character that has no storyline significance outside of the game's tutorial mode. Eliwood and Hector were basically the real heroes of FE7.
Micaiah: One-shot Fire Emblem character that gets upstaged by Ike about 1/3 into her game (though not to the extent that Lyn get overshadowed in FE7).
Eirika: One-shot Fire Emblem character that shares her game with her more popular twin brother who is also a lord. Does not help that this game is the least popular of the GBA Fire Emblem trilogy.
While both Lyn and Micaiah have only appeared in one game each, they are representative of major recurring archetypes that have appeared in almost
every Fire Emblem game. It is no secret that the Fire Emblem series leans heavily on a set of major recurring archetypes in order to build the cast of each game. For example, every Fire Emblem game has a Red Knight, Green Knight pair of two cavaliers who join early and have opposite personalities.
The reason that Ike is an interesting Lord character is because he draws upon one of the major recurring archetypes other than the classic Lord archetype: the strong mercenary, much like the original Ogma. Drawing upon that archetype is the reason that he plays so differently from Marth in Smash Bros. Adding him in also brought a slightly better reflection of the variety of characters from Fire Emblem than what two traditional Lords could bring.
Micaiah is just the most recent incarnation of the Female Light Mage archetype that has been around since Linde in the original Fire Emblem. The archetype really rose to prominence with Yuria in Genealogy of Holy War, who was so important that you can't beat the final boss without her. Arguably, the archetype also includes Lilina, daughter of Hector and secondary main character of Sword of Seals. Micaiah draws upon the history of that archetype and Yuria in particular, being a light mage whose powers are key to defeating the most powerful enemies in the game and whose identity is central to the plot.
Lyn is a good choice for Smash Bros. because she represents the opposing myrmidon style of sword-user to Ike's mercenary style. Female swordfighters are pretty common across the Fire Emblem series and often good. Lyn in particular gets credit from having what are probably the best critical hit animations in the entire series.
There are at least two other really good options for female characters for Smash Bros. from the Fire Emblem series. The first is the recurring Pegasus Knight Princess character, which started with Caeda from the original games (who is necessary for more character recruitment than Marth himself) and continues all the way through to Elincia from both Radiance games. Both Caeda and Elincia are just one step short of being full Lord characters, and both can be very useful.
The last possible character is the dragon princess Tiki, or one of the other various "royal member of a clan of people who transform into Dragons" characters, like Kurth, Nasir, and Ena from the Radiance games, or Myrrh from Sacred Stones (for the purposes of this discussion, Kurth or Nasir might not be the best pick, but in a more general sense they could work as a Fire Emblem representative, same goes for Dark Dragon Medius).
If any Nintendo property is going to produce a good candidate for more female characters for Smash Bros., it is going to be Fire Emblem. There have been far too many good female characters from just about every Fire Emblem game to reject the idea outright.
Furthermore, it is important when representing the Fire Emblem series to look past just the main characters. Most of the main characters other than Ike have been clones of Marth. Focusing just on "main characters" of the Fire Emblem series would be like replacing Zelda and Ganondorf with two more variations of Link, or replacing Luigi, Peach, and Bowser with Paper Mario, Doctor Mario, and Tanooki Mario. Smash Bros. works better when it represents the breadth of a property rather than just the absolute main characters.
And I have to add that it feels weird saying all of this in response to someone with an Emperor Alvis avatar...