I might as well give my two cents.
ME1:
-Start the story with Shepard already a captain, somewhere in his/her early to mid thirties
-Spectre status is something only given for a limited time to soldiers so that they may act as Council agents for a mission. After completion, Spectre status has ended, and the agent goes back to his/her military. This way we keep Shepard in the Alliance chain of command.
-Garrus and Wrex would be mandatory squadmates, just for the sake of plot consistency in later games.
-Make Virmire only available after Therum, Feros and Noveria; that way writing can build up to the big reveals on Virmire without worrying about whether or not the players have finished the other stages.
-Give Liara a justified backstory for why she's a combatant (justicar/huntress/etc)
-Give Liara an actual reaction to her mother's death
-Have an NPC take over the role of Prothean expert. Personally, I would find it interesting to get a Hanar perspective.
-Give Garrus a more interesting history than "alien cowboy cop". Perhaps explore and deconstruct the idea of Turian honour and absolute obedience.
-Give every character a clear character arc so it actually feels like this journey has seen them grow.
-Rachni Queen decision becomes to either let her go, or to leave her to the Council.
-Have Kirrahe get off Virmire alive no matter what (more on this later)
-Have Vigil give you the blueprints of the Citadel as well as all the data the Prothean scientists had collected
-Explain that the Prothean scientists severely sabotaged the Citadel's ability to control relays, so this doesn't become a headache in ME3
ME2:
-Skip the whole death/resurrection schtick. Put a 5 year gap between this and ME1, and explain that it was spent fighting Saren's Geth and decoding all the information Vigil gave.
-Have the Council-Geth War still raging on. Thus the Council can't help when human colonies are disappearing, so Shepard turns to Cerberus out of desperation.
-Cerberus still has the backstory of an Alliance black ops group gone rogue. Except that's only the official story so the Alliance can maintain a clean facade with the Council. In truth, Cerberus, now "freelance" under the Illusive Man, is still very much the "left hand" of the Alliance.
-Make all ten core squadmates mandatory
-Amalgamate Samara and Liara's characters (since Liara is already a warrior in this continuity) and role as a squadmate
-Axe the recruitment missions for Garrus, Tali and Liara. They can join you as soon as you get the Normandy SR2
-Now that three of these missions are gone, put those resources into making Lair of the Shadow Broker part of the main game
-Kirrahe becomes the Shadow Broker, since he actually has a background in information and intelligence
-Make the ten core loyalty missions mandatory. Give them all a premise that ties in (at least somewhat) with the main plot.
-From the previous game, if the Rachni Queen was let go, then she sends you a thank-you message. If you gave her to the Council, then you get reports of her being experimented on
-Tie Grunt and Okeer into the Rachni somehow. Perhaps Grunt is made with Rachni technology instead of Collector tech?
-Make Jacob a foil of Shepard's, a battle hardened soldier. Left Alliance because of disillusionment. Have his loyalty mission focus on his history as a corsair. Have a key character trait of his be a belief that it's better to be a conqueror over a destroyer (this will come back later).
-Make the Suicide Mission an all-or-nothing scenario. Either everyone dies (fail ending), or everyone lives (success ending)
-Keep Chris L'Etoille's ideas on destructive analysis of species forming the Reaper neural network
-Change the "humans are genetically diverse" to "humans are not genetically diverse but still show a wide range of phenotypes like other sapient races hence they're viable to be turned into a Reaper mumbo jumbo"
-Have Shepard keep the Collector base no matter what. Emphasize that by preserving the base, they now have the blueprints of a reaper, giving them an unprecedented level of insight into the enemy that past cycles lacked.
-Change the ending scenario to destroying the Collector breeding facilities on the base, or preserving them so Cerberus can breed and control a Collector army for the Reaper War
-Have the ending hint that Harbinger is having a second Reaper constructed with the Batarians
-Scale back Harbinger's taunts, and make the possessed Collector something akin to the Praetorian in terms of toughness.
I'd split ME3 into two games, simply so all the story arcs can be done justice without rushing it all.
ME3 and ME4:
-Have more time pass between ME2 and ME3. Explain that in the years in-between, the Alliance in collusion with Cerberus, the STG and the Shadow Broker began to draw plans for a weapon that combined the knowledge of Reapers from the Collector Base and the data Vigil gave on Illos. This weapon, codenamed the Crucible, targets Reapers by linking up with the Citadel and unleashing its energy via the relay network it controls
-Because every squadmate from ME2 is alive, they all become part of the squad at different sections of the game, rotating in and out
-Scrap Vega and EDI, Have a Batarian squaddie.
-The opening mission is the fall of the Citadel. C-Sec is indoctrinated and sabotages the station from the inside, the Batarians and their Reaper larva attack from the outside. The larva links to the Citadel, and succeeds where Sovereign failed. The Council manages to escape with the remnants of the Citadel fleets, and Shepard, now a Rear Admiral, rushes to Earth to warn the Alliance.
-The Reapers pour through the Citadel and split into three. A small faction (along with the allied Batarians) hits earth to prepare for the harvest, while the majority go and open two fronts against the Asari and the Turians.
-The Reapers, while all individually very powerful, are very low in numbers due to how picky they are over which species ascend into capital ships. Scrap the concept of Destroyers.
-With the vast majority of Alliance leadership dead in the initial Reaper assault on Earth, Shepard ends up only a few ranks away from the top. Admiral Hackett is still the one leading the Alliance.
-Cerberus becomes an enemy-of-my-enemy-is-my-friend type ally of the Council races, "generously" offering its easily manufactured ground troops in exchange for an increased voice in Council politics. There will be times where you work alongside Cerberus, and times where you disagree with them and fight. They still strongly advocate controlling the Reapers.
-Thessia becomes the new seat of the Council, and a hubworld so we can care for it when it falls.
-Wrex demands a cure, but only for 1/100th to 1/50th of the Krogan fertility rate. This shows that he's got the brains to know he can't control a massive Krogan boom while trying to be a reformist. However, the other Council races are still opposed to it because it's a huge boost to Krogan fertility.
-Shepard isn't the one making the decision about the cure. At the Shroud, Garrus and Mordin choose to either dispense the cure, sabotage it, or betray one another depending on how you handled their respective loyalty missions.
-Similarly, Rannoch is resolved solely by Tali and Legion, again depending on how their loyalty missions were handled. Rannoch gets three conclusions after the Reaper is beaten: Peace (Geth reject the Reaper upgrades), Control (Xen subjugates the Geth) and Destroy (Garrel destroys the Geth).
-The Geth don't get whitewashed, and the Quarians don't get villanized. Peace also is not the optimal choice from a pragmatic point of view because the Geth reject the Reaper upgrades and are left weaker, whereas with Control their vast numbers are preserved at the loss of their freewill.
-Have a meaty story arc with Grunt revolving around the Rachni. The Reapers have their own Rachni Queen, but if you spared the Queen from ME1, she allies with you and helps destroy the Reaper doppelganger. If you left the Queen with the Council, she reluctantly aids you, but in more indirect ways.
-ME3 ends with a Krogan-Turian-Qurian (+Geth+Rachni) assault on liberating Palaven
-ME4's first act revolves around bringing down the Batarian Hegemony. Depending on how it's played, either they become your allies or are completely destroyed in the conquest
-Thessia falls if you have the new Asari Councillor that doesn't fully trust you. If you saved the DA in ME1, the old Asari Councillor has enough pull to get the majority of the Asari to retreat before Thessia is taken.
-Omega becomes a core part of the game, and involves a decision to either leave the station with Aria and let it go back to its old ways, or leaving it with Cerberus and giving humanity a strong foothold in the Terminus.
-You discover Miranda is behind Sanctuary, experimenting on prisoners that Council races no longer have the resources to hold. Sanctuary is less stupid-evil horror and more necessary evil.
-Jacob betrays you at some point, being indoctrinated, believing he can Control the Reapers himself
-TIM also betrays you, but only pretends to be indoctrinated so he can sneak on to the Citadel and modify the Crucible for Control. However, a key difference between TIM and Jacob is that TIM intends to have EDI control the Reapers as she's made from Sovereign's neural network.
-Miranda can be convinced to take over Cerberus after TIM has fallen.
-By the endgame, even Hackett falls to indoctrination, wiping out another good chunk of Alliance forces and leadership. Shepard ends up the highest ranking Alliance officer left alive and thus stands as an equal to other galactic leaders at the final assault.
-Earth only becomes important when the Crucible is moved to it. The battle of Earth consists of multiple cities and levels, and war assets participate in battle
-No holokid. The ending simply revolves around the ideologies of Control and Destroy.
-Picking Destroy disables the Reaper FTL drives, leaving them stranded above Earth and forcing them into a bogged down fight with the galactic alliance. Leaves Earth torched and humanity nearly extinct.
-Picking Control has EDI seize control of the Reapers. Leaves Earth intact, but the threat of Reapers still hovers above the galaxy for the rest of time.
-Shepard's death dependent on multiple factors. Can get a happily ever after if played right.
-All past surviving squaddies return for final mission