Something drastic has to happen

I'd make the next game completely about the Quarian ark. Forget all the generic Kett and Angara stuff, there's no way you can make that lazy piece of garbage exciting.

I would tell the story about the Quarians who arrive in a completely different part of Andromeda. One that is much more hostile and dangerous, especially considering the vulnerability of Quarians to such things. The Quarian Ark gets damaged on arrival and is unable to continue flying, rendering it immobile.

You play as a Quarian Engineer/soldier/scientist or a different option, which is both your class and your occupation in the story and you have to go with a scout team to try and find materials from the closest planet that could be used to fix the ship and get it moving again. The planet you land on is harsh in conditions, you encounter no alien life but the natural environment is dangerous enough to kill several on your team, and nearly the Quarian you're playing as.

You survive however, and recover materials that seem to be familiar enough to be worked into usable parts that can fix the ship. The ship gets moving again but you uncover a terrible truth. Along with the materials you brought came a pathogen that spreads and infects Quarians, killing them outright... or creating something even worse (not exactly Necromorphs obviously but something close to that, inspired by Dead Space, I know I'm unoriginal).

Anyway the next part of the game would be you trying to survive on this ship, helping as many others as you could find and making sure that it stays afloat and doesn't plummet into the nearest planet (a bigger, much more advanced version of the Joker mission in ME2). Eventually though you see that if you stay, it's only a matter of time until you yourself become infected so you have to escape the ship.

You disable the ship once again, to make sure it stays where it is, board an escape shuttle and set out in the star system to try and find something that could help you fix this situation and save what's left to be saved. You're with a few Quarians that you saved and perhaps a Geth unit that was snuck aboard by someone, thus creating the squad.

You start going to the nearby planets and this is where the game opens up, you seek allies, resources, perhaps a cure of some sort. Each planet offers different challenges though and the environment is the harshest enemy of all, but eventually you do of course run into wildlife in Andromeda, and finally another species as well. But the species is not humanoid or English speaking, in fact it is so alien and so different from you that you spend most of the game even trying to figure out how to communicate with these things and how you approach that, is up to you as the player.

I think that this version of seeing the Andromeda galaxy would offer a much more compelling and intense experience. Sure, it may be a bit tougher to relate to as you're playing as an alien but I feel like it would be very possible to humanize the Quarians and make you see the dynamics of their culture and relationships at the start of the game, making you care for them. It would be a bit more linear in places and there would be less shooting and more just navigating the strange worlds your visiting. The communication with another species would make for a unique experience as you see what you're really like - are you patient and attempt to figure out the best way to communicate with someone or will you try to seize control over them and essentially make them your servants. There's a lot of possibility for narrative choices and some new and experimental themes to be explored in the Mass Effect universe.