The single biggest change I'd make would be to drop Shepherd as the reoccuring main character.
A lot of the tensions and contortions that went around the ME trilogy were because of its often nonsensical focus on Shepherd in particular. The only game in which Shepherd has a unique ability or trait apart from being the PC was ME1, when Shepherd is the living holder of the vision mcguffin. That, more than anything else, was why Shepherd was important- and it was utterly irrelevant in the grand scheme of ME2 (where everyone, even the villains, are Shepherd fanboys) and ME3 (where Shepherd is Hackett's gopher).
Because of the reliance on Shepherd, the story had to focus on tying characters back to Shepherd. Cames, especially crew, were dictated by 'bringing back the gang' even when it didn't make much sense. Companion characters went through absurd, nearly nepotistic, career paths to justify them being around and relevant for Shepherd once again. Tali goes from 'not a legal adult' to 'admiral' over about three years. Garrus quits two jobs to become someone Generals salute and say 'sir' to, only to quit that one to become a side kick. Jack, somehow, is considered quality teacher material. Etc.
Shepard could have easily been replaced without issue, even when keeping most of the same cast. If ME2 were, say, the story of player-character Jacob/Janett Taylor (customizable, of course), leading a Cerberus cell on a suicide mission, that would have changed... not a whole lot. Nearly all the companions of ME2 join for reasons utterly unremoved from Shepherd. Those that did have Shepard-specific motivations (Tali, Garrus, Legion) could pretty easily have been written to have Cerberus-compatible recruitment rationals (Tali to spy on Cerberus for the fleet, Garrus maybe/kinda as a quasi-Spectre, Legion trying to open up back-channel relations to human authorities through the Cerberus cabal).
Vega (again, customizable PC) could have been just as responsible for the ME3 war hero arc as Shepard. Ignore the token rationalization for why Shepherd is needed for Surkesh ('we need a spectre!' 'oh, and have your spectre status back regardless, so that we can need you'), and Vega could represent Hackett who represents the Alliance as the neutral party for the war conference. PC magics allows the victory on Tuchanka, creating the alliance and an instant war hero. War hero goes on to save Rannoch, do great things, but also have the guilt complex dating back to the Collector Abductions and current events. You could even write the companion crew back in, with an angle of 'it's the Virmire Survivor's ship and crew, with Vega steps into and takes over after the VS-mentor is wounded.'
So, yeah. Getting rid of the Shepherd focus is my big one.
Otherwise...
ME1 needed backstory/buildup changes to somehow justify how Humans were relevant and/or disruptive. Either the Human backstory needed to be expanded with a longer buildup/colonization period before contact, or something else. As it is, there's no credible reason for the ME galaxy to consider Humans a major power... if the Humans had anything to justify being a power in the first place. This dynamic is very important to the ME1 question of Human-Council relations.
(Possible solution: have the entire galaxy under a sort of universal-genophage regime, in which all species are kept to a Council (read: Asari) standard fertility rate so that another Krogan Rebellion/fast breeding ascendant species is impossible. The Council species have lower genophage burden (the Asari are the standard, the Salarians keep non-genophage breeding controls, the Turians are the canon fodder), which is a grievance. But, since a major war is almost unthinkable because the population loss would be a major blow for whoever fought it, the genophage is considered a requirement for galactic peace. A major Human/Council tension is the prospect that Humans, with a birth rate higher than the Asari/galactic standard, are negotiating/may be forced to endure a genophage as part of galactic integration. Renegade-ism ties into the 'we won't let them neuter us' defense, while Paragonism ties into the 'we should shoulder the burden everyone else does for galactic stability..))
ME2...
Regardless of whether we keep the Reaper plotline, it needed focus on the Council-Terminus cold war and Batarian Rebellion subplot.
If we keep the the Reapers, the goal of the Collectors should have been to prepare the groundwork for the Reaper invasion by setting everyone against eachother. Rather than abductions to build a Reaper, the Collectors should have been staging abductions so that the Humans would blame and attack the Batarians for slave abductions. In addition to a Human-Batarian war, the Collectors could have been involved in the Geth Civil War/sabotaging Geth/Quarian peace efforts, as well as in manufacturing a genophage cure to start a new Krogan Rebellion. ME2's end would have prevented an imminent Council-Terminus galactic war, but set the fires for other conflicts to be resolved in ME3.
If we ditch the Reaper apocalypse angle, then I wouldn't mind ME2 being a Cerberus-centric plot of advancing human interests from a different perspective. Instead of Collector Abductions, the plot idea around the Suicide Mission could have been 'the heist': the Collector Base is a known/suspected Prothean archive, untouched and intact and considered impossible to get. Considering what a single Prothean beacon is worth, and how it was feared that the Eden Prime beacon could set off a Council-Terminus war, the gains for reaching the base could be galaxy-shaping. Cerberus wants it for itself, and inbetween advancing other Human interests the Cerberus cell is putting together the team and resources to crack this. The Cerberus leader in charge of this Suicide Mission needs the best of the best, but also needs to earn the loyalty of those involved so that obvious betrayals are avoided and that no one sabotages the big prize. (As an alternative to loyalty = survival, now disloyalty = sabotaging the mission.) Once the Collector Base/Prothean Archive is cracked, Cerberus finds its own Prothean dreadnaughts (read: Sovereigns) and other such toys, just as a galactic conflict is sparking.
ME3:
Going down the Reaper plotline... aside from starting Rannoch from scratch, I'd probably make the Reaper motivations fit the Dark Energy theory. Or a variant of it, anyway: dark energy degrades with time, but civilizations build it up too fast. Reapers create the Reaping system to centralize and make it easier to wipe out civilizations in a controlled manner. If they didn't, species would still naturally discover/invent e-zero on their own, which would make the problem even harder to root out. The Crucible would become a massive e-zero engine which could either destroy all e-zero in the galaxy (including the Reapers, though this is revealed later as a secondary effect), or trigger the dark energy inflection point of no return.
Destroying e-zero is basically the low-EMS destroy ending of canon, in which galactic civilization is sundered. Alternative FTL technology is possible (and would be captured by Cerberus in the Collector Base if the Base is kept), and mass effect can eventually be re-created, but the dark energy problem is reset and mass relays are destroyed. Shepherd, if lucky and alive, can make it down to Earth before the effect hits, letting them live with the LI (though the dextro-aminos are fucked) in the time they have left.
Alternatively, the player could call the Reaper bluff and kick off the dark energy inflection point. This dooms the galaxy to heat death... in a few thousand/ten-thousand years. The Reapers, not seeing any logic in staying behind and wasting resources on a doomed cause, depart to 'protect' other, less suicidal galaxies. The survivors in this galaxy have a chance to prepare for a trans-galactic exodus where they can live more sustainably in the next galaxy, or to see if they can figure out what the Reapers could not about preserving/reseeding this galaxy with stars. Depending on Council choice and other Big Decisions (Geth vs. Quarians especially for mega-structure vs. migrant fleet help), the future either seems doomed to selfish self-interest as species fight for resources to use to escape, or a cautiously optimistic one in which everyone comes together to work on a solution and leave no species behind. Shepherd lives, and gets to live with the LI in the twilight of galactic civilization.
The third way, for those with enough War Assets and stuff to keep the Reapers at bay, is to basically play the Crucible as a bluff to make the Reapers back down. Either function of the Crucible is a 'loss' for the Reapers, but also a loss for the galaxy. Shepherd poses a deal in which the only way to win is to not play: either the Reapers stand down and accept a cease fire, allowing both sides to work on a solution for the Dark Energy problem (Paragon), or Shepherd fires up the Crucible (the Renegade threat). With enough credibility and reputation, this works. What emerges is an incredibly unstable and uncertain cease-fire: Council/coalition forces protect the Crucible, Reapers doubtless planning on how they can take it out and resume their harvest. But the sides are also breaking down: not just the species, who have no consensus on how they might address the element-zero reductions necessary to avoid the Dark Energy problem even as they recoup and rearm to resist the Reapers in a Round Two, but amongst the Reapers, who fragment into different factions on how best to proceed. Some reapers cooperate, some conspire, and so on. No one knows how long this cease fire can last, if it will last, and if the dark energy problem can actually be solved. But there's a chance, and it just might work, and whether it does or not Shepherd is a living legend as the person who stared down the perpetrators of countless genocides and made them blink.