I don't think scientific impossibilities are a bad thing at all. At the end of the day, this is science fiction. Of course it's going to have nonsensical elements that just can't work in real life. Realism would mean we play as a NASA bureaucrat who tries to get another drone deployed to Mars in 10 years or something. It's just not fun. Until we actually leave our planet and/or encounter aliens (which isn't likely in the foreseeable future), any science-fiction story relies on convenient nonsense by defintion.
That said, I do think there's a difference between scientific mumbo-jumbo that was introduced early in the setting and makes some amount of sense when presented to a layman (the titular Mass Effect, for instance) and something that is just so mind-boggingly ridiculous that thinking about it for a minute makes the entire concept fall apart even with no proper scientific knowledge (such as the Synthesis beam). The first preserves the suspension of disbelief. The latter does not.
In a universe with magic, this means that a mage throwing a fireball is OK if it was explained it's a common magic power, but introducing a character that can make the entire planet love each other with a snap of their finger with no explanantion is just cheap. Neither is more realistic than the other, yet the audience can learn to accept the fireballs, but probably not the wave of instant peace and love.
So yeah, I disagree with In Exile that it's all just space magic nonsense with no distinction. Building a credible world, or at least a world with a convincing enough veneer of credibility, is important to writing science-fiction that I enjoy. For all its faults (Project Lazarus, urgh) and missed details, I thought Mass Effect did a good enough job of that until the ending made the entire thing crash down.
And yeah, I'd like to have an explanation on how they managed to get to Andromeda with no need to discharge their drive core. If what Mac Walters let slip by in a recent interview is real, the journey took 600 years. That's actually somewhat consistent with the speed of FTL as described in the original trilogy. But where did they discharge their cores, is a question that kinda needs answering IMO. It's a rule of the setting. Ignoring it now isn't a great idea, no more than it would make sense for all the mages in Dragon Ages to all be able to turn into dragons at will with no explanation given despite the fact that magic, by its very nature, is nonsensical.
There are a few points to address. First, I disagree that nonsense introducing early on is better than nonsense introducing later on just because it's easier to bake into the introduction to the setting, but I think we're all talking past each other.
Second - there are two streams here. The first is that when we ask whethere a new idea is nonsense by the standard of the setting we have to actually look at what the setting introduced. Synthesis - as discussed below - is no less stupid than anything introduced in any part of ME. So the complaint that it "violates" the rules of the setting is misplaced - the setting always had such ridiculous ideas.
The second stream is actually about internal consistency. But that's really up for debate. Take thermal clips - this ridiculous idea has an in story justification that's no less stupid than many of the other ideas in ME1.
When I'm talking about ME using nonsense, I mean that the series has established a pretty flagrant disregard for how scientific progress actually works and the rules of their setting are very fluid.
This drive core nonsense is not an argument about internal consistency - it's an argument about science. If driven cores suddenly didn't require discharges - that would be a bald retcon. But we can have actual justifications - like the ridiculous thermal clips. That has an in-setting justification (for a gameplay mechanic).
I don't think Synthesis is actually an issue at all. Because it's no less mind numbingly stupid that psychic plants, psychic blue space babes, and gravity superpowers.
Let's use your magic example. Why would having such awesome magical power at all not be consistent with the setting? Are nuclear weapons "lore" breaking because they were a hitherto unknown way of waging war with awesome destructive power until we invented them?
All that you're saying is that the setting has to offer some thin explanation for why something is happen -but settings often to that part. Again, thermal clips.
The debate isn't really about consistency - it's about what people what the setting to look like, with things like "consistency" being silly appeals to justify their real position, which is that they simply don't like the change.
When I bring up the stupid nonsense and ignorance of science in ME1, I don't mean it as an indictment of the series as much as an illustration that the series IS being consistent when it introduces an idea that (1) ignores how scientific developmentb works (2) would be comically uneconomical and a misapplication of the technology if it did exist and (3) would raise all sorts of questions about why it wasn't used before. That's all this driven core discharge stuff boils down to - and it's a complaint about science because it's really an argument that the Citadel races could not actually develop this tech.