The fictional by the time Mass Effect begins in 2183, they still rarely live much longer than a century and a half - little more than a blink on the galactic scale. While Commander Shepard was able to save the galaxy from the Reapers in roughly four years, most galactic occurrences don't happen so fast.
Humans are seen as ambitious and impatient, partially due to their short lifespans. Other longer-living species in the Milky Way, particularly the Asari, have a completely different perspective on the interstellar community. Capable of living for over a millennia, the Asari are often considered the "most advanced" race in Citadel space. Though most aren't particularly smug about it, the Asari were the first to discover the Citadel, and are the progenitors of the modern interstellar civilization.
A new Mass Effect game with people playing as an Asari could potentially be radically different than the four games currently in the series. So far, all three games in the original trilogy and Mass Effect: Andromeda have featured human playable characters. Asari characters were available in the Mass Effect 3 multiplayer, but that didn't offer the series' traditional narrative RPG experience that a whole new Mass Effect game centered on an Asari could.
Playing As An Asari In A New Mass Effect Game
Creating a new Mass Effect game revolving around an Asari protagonist would probably be a massive undertaking. The long lives of the Asari are so integral to their personalities and culture that it would have to be featured in a game's story in order for the narrative to be effective. This means creating a game that spans centuries, with conflicts that last lifetimes. The Mass Effect universe has an extensive history, but creating specific events that have their individual ramifications over a thousand years would be an incredible feat.
Time jumps would almost certainly have to be a necessity, but that may leave this new Mass Effect game feeling disted. However, the Asari lifespan does nicely align with a three act narrative, and could also be used as the framework for character progression. Liara tells Commander Shepard that she is barely considered an adult at the age of 106, when she is considered a Maiden in Asari culture. Maiden become Matrons around 350 years old, and Matriarchs near 700.
There are already compelling narrative impetuses in the existing Mass Effect games for one centering on an Asari protagonist. Becoming a Justicar - a sort of knightly version of the Council's Spectres - like Mass Effect 2 and 3's Samara would allow the player to seek out wrongdoing, offering plenty of possible story beats. Another fascinating route would be to make the player an Ardat-Yakshi, an Asari with a genetic defect resulting from the mating of two Asari - exemplified by Morinth in the games. Making the player an Ardat-Yakshi would provide the game a very dark tone, since they are often exiled. An Ardat-Yakshi has the possibility to kill their mates while melding, becoming stronger from the experience.
The Asari are one of the more prominently featured races in Mass Effect, and would make for a compelling protagonist. There are already interesting narrative possibilities within their established culture, but creating a game around such long-living beings could prove difficult. Mass Effect has so far not allowed single player protagonists that aren't Human, but the Asari could be a compelling alien race to focus on.