I suspect this thread has devolved into something else, but to give my thoughts before trecking through a dozen pages...
I think the single biggest thing I would have done would have been to largely nix reoccurring companions. Pretty much every character arc that went past its initial game went from bizarre to despised to nowhere at all. Character development arcs became schizophrenic as they bounced around or forgot what they already established, and especially in ME3 a number of the companions were shoehorned into not exactly credible places just to be present.
A good companion cameo is more than just a moment of badass and some 'just like old times' lines, complete with buddy-buddy moments. Wrex in ME3 is an awesome role as the Krogan leader... if you don't pay attention to Wrex's philosophy in ME2. Thane was in a great position... unless you listen to the Thanemancers and ask their opinion. And as for Admiral Tali, 'Generals salute him' Garrus, or Shadow Broker Liara?
Well, I'll just say that it wasn't a natural evolution from where they were in ME1. Granted, ME1 Tali was a chickenleg codex, but I miss the Garrus who had a character arc about maturing and getting out from under Shepherd's wing.
Forcing companions into major roles might have been popular, but I don't think it was good writing. I'd have a much stronger of Tali's character arc in ME2, for example, if it had been 'she's on the team for politics' rather than trying to retroactively justify her as a 'best of the best' in the galaxy... while providing little justification for that. Let her character arc be realizing that she isn't as good as everyone else wants her to be... but that she can still be good enough.
(My thoughts on Shepherd as the protagonist on the trilogy are long established. I see little benefit and more weaknesses from the insistence on him/her.)
Now, for a short list per game?
ME1: A good case of character arcs that got dropped when characters were shoehorned into later games. The primary non-offenders are Wrex and Tali, for different reasons.
Ash/Kaiden- I would have made it so that you didn't have to be the opposite gender to Paragonize Ash/Renegade-ize Kaiden. And I wish they could have kept those alignments- the VS got a lot more bland, and lost a crux of their ME1 character arc, when the ME1 development got dropped.
Garrus- Why did ME1 Garrus have to die? His characterization arc, one of the strongest in the game (not least because you didn't have to be a potential love interest to trigger it), was simple, compelling, and had a clear endstate. Garrus learns patience and responsibility and goes back to C-SEC, or Garrus affirms his desire to do good overruling the bureaucracy and goes back for Spectre. And then... poof. It's all thrown away, rather than utilized, to throw him back under Shepherd. And then, after becoming a Big Person and having such responsibility that generals salute him, come ME3 the serial quitter... quits again, and not a single Turian tries to stop him or expresses regret that he's leaving. There was an intent with that planet, and it rather backfired for me.
Liara- Never needed to become the Shadow Broker or try a Dark and Edgy route. All her relevance to the core plot, and then some, could have been kept through her established history as a Prothean/Reaper Cycle archeological expert.
Wrex- already mentioned, but ME2!Wrex had a compelling governmental philosophy for the Krogan. ME3!Wrex ditched that to rely more on ME1.
Tali- could have had actual characterization. Personally I think 'is treated special because of her political status, but isn't suited for the responsibility (yet)' would make a better character arc than what they tried to make ME2 about, and would actually lead into Admiral Tali, but whatever was done should have started in ME1.
ME2- well, the cast was way too large in my view, precluding much interaction and exaggerating the lack of relevance to the main plot. Pretty much everyone could have used some tie-in to the Collector threat... if you didn't chop the cast as a whole. You could probably combine almost half the companions into composite characters, and get stronger characters as a result- Jack/Miranda, the stockholm syndrome victim/advocate of Cerberus, Jacob/Zaeed an old badass who's seen all the galaxy has to offer and yet still has a core of morality under a suit of sour armor, and even Samara/Morinth (I'll let you imagine the possibilities of an AY trying to hide and live as a Justicar).
There are really too many things I would twist about the companions, so I'll bring some points about cameo encounters from ME1.
Lorik, the Turian from Noveria who you could get a garage pass from, would have been an excellent replacement/substitue for what's-her-name if you didn't side with her. His presence on Illium would be easy to justify (business), his role as a quest giver similar (you helped me last time, perhaps you'd like to again), and in general he is an unkillable substitute for what's-her-name.
Shiala, from Feros. Rather than have a nameless substitute if she wasn't around, use the unkillable Sarah from the same encounter. Sarah was the Exo-Geni employee you found who provided data on the Thorian. In addition to being unkillable herself, she could have a similar basis for wanting to help the colonists (guilt), has a basis for being the agent on Illium (experience with corporate red tape), and otherwise has as much of a reason for a hero-crush on Shepherd as Shiala.
The Rachni Ambassador could have been replaced with a Noveria executive or Krogan who had heard about the Rachni destruction. Rather than no encounter at all, you could have someone who knows thank you for destroying the Rachni and off a little thank-you gift (and immediate, but smaller, reward than Rachni assets come ME3).
Fist. I think it would have been hilarious if, assuming you killed Fist, if you sat down at Afterlife his son tried to confront you for revenge. It'd be all like 'Hello, my name is Inigo Fist. You killed my father. Prepare to die-' bam. Cue Shepherd's teammate, lowering gun, going 'who was that clown?'
Oh, also, the replacement Council could have shown up. And Udina didn't need to be degraded to needless jerk, compared to ME1 where he was an ass but an ass who would agree with you.
ME3- well, I've written my own interpretations of the game in my Story Corner, see my signature block. Key points?
-EDI didn't need to be a sex bot to have a character arc... though I would have laughed had 'she' been put inside a male chasis and given Joker quite an identity crisis.
-The Quarian Admiralty board didn't need to be reduced to idiots, and the Geth so one-sidedly whitewashed.
-Someone could have challenged Wrex on how he intended to hold the Krogan together peacefully if the genophage was cured.
-TIM doesn't need to be indoctrinated to be an antagonist. He doesn't even need the whole 'Control' motivation, which wasn't really developed enough to be warranted in my view. Cerberus could just as well have been playing the long game with a post-Reaper perspective, trying to prevent the rise of any new superpowers (genophaged Krogan) and helping the Reapers to level the opposition while banking on the Crucible gambit.
-Kai Leng is lame, and could just as well be replaced with an exceptional Cerberus mook. (Fun thought- imagine it was a super-Centurian, and when you finally beat them you take off the mask to reveal an opposite Male/FemShep.