Hm, apart from the fact that I know from previous conversations that we very much disagree on what contrivances ME1 had exactly, I do think "following up on" and "patching handwaves" do go hand in hand. Unfortunately, even if you want to separate those two, ME2/3 did neither very well.
If they were to make a new trilogy in Andromeda, I would very much hope for substantial improvement in that particular area.
It depends on how you define "planned". Yes, obviously they had an idea that they wanted to make 3 games but IMO, planning a trilogy means exactly that you have an idea of where you are going before you get to the third part.
As for re-inventing the plot: IMO, the plot of the trilogy does not feel very coherent or even continuous. ME2 especially goes to great length to confront you with problems that have nothing to do with the main premise of the trilogy and therefore stalls the plot there. ME3 tries to get the ME1 plotlines back on track but in doing so, it again breaks with what little progress did happen in ME2. Ultimately, while all 3 games in themselves are very good at presenting and advancing their own plotlines, they just don't fit together very well as a coherent whole (which is what I expect from a trilogy).
Well, the plot is more or less the same, Reapers are coming, how do we stop them. That basic plot thread is present in all three games. What I think you are describing is the narrative of all three games instead, which are all different by design.
The narrative focus is why they are disjointed a bit. The first game is the road movie, the second game is the dirty dozen, and the third game is the war movie, essentially. So game one was tightly focused on finding a rogue operative and exploring the galaxy, game two was all about interpersonal relationships and setting up bits of the universe. The galaxy is lived in now so you don't need to find every nook and cranny again, while game 3 is the climax, the big fight that has been on a simmer since game one. A lot of that game is consequence over choice, whereas the first two games were a lot of choices with less consequence.
Pacing issues in all three games aside, which they all have a lot of, most of what you see in the games are NOT supposed to be a part of the main plot in terms of narrative structure. Got to remember, the first game is about Saren, not the Reapers, who are more or less shown to be the true villains in act 3. So much of what you are doing in the first game, the side-stuff and tracking Saren, is either narrative-related so it can accomodate choices, or slightly plot related regardless of said choices.
The second game is consciously tied to interpersonal relationships, and many of those conflicts come back in game 3, especially for characters like Mordin, Tali, Samara, Jack and Legion. It doesn't have to be married to the plot per-se, but it does have to give narrative context to why you are doing these missions; to gain trust and make sure you survive. It's not interested in confronting you every moment of the day with the plot, but it also doesn't have to either because of the narrative.
Game 3, being the climax, is the closest-tied to the actual plot if you really think about it. Yet the biggest missions and moments only have some degree of plot undertone to them, it is the narrative choices found in the first two games that dramatically change how they play and feel in game 3. Tuchanka and Rannoch obviously come to mind; yes it's tied to the plot because were recruiting, but it's in the background, much like Mass Effect 2 and how gaining your crews trust and loyalty is tied to the plot of assembling a team to assault their base. It is an insignificant connection, but it is present, and like I said, the strength of the series is its not married to the plot all the time.
If it was, Citadel and other DLC would not exist.