Not sure if first person would be a good idea, but I agree that she needs more depth.
Yes, Gj Riot for releasing a nonpandering female monster champ, but at the same time it would be incredibly boring for her to simply be a monster based on instinct.
IMO, the overall plot increases in depth when the "monsters" have some explanation. For example:
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The rachni from mass effect are initially perceived as mindless monsters, but then you learn that they are actually very intelligent sentient creatures who communicate by songs.
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The darkspawn from Dragon Age are initially presented as unknown mindless plagues, but their origin turns out to be far more messed up. Throw in the awakening DLC plot in there for and extra good story time.
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The zerg overmind is/was a very charismatic personality; without it, the race would simply not be as interesting
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The witches in the Madoka anime
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Song of Saya visual novel
Of course, it depends on the context. In some cases, details can ruin the experience (Reapers explanation in ME3)
Rek'Sai is a nonspeaking monster champ placed into a setting that currently doesn't exist because Riot deleted the old lore and didn't make the new one yet. There is no framework or context, no speech, no dialogue, NOTHING. Broodmother in Dota2 has a voice, and it's cool; she doesn't lose her monster card just because she's voiced. Giving Rek'Sai minor personification or some context isn't going to diminish her status as a monster, and more than likely may strengthen it.