Again, literal space magic. There is a difference between an exploited mineral and a sorcerer's spells.
Not when the limitations of said mineral are unknown, especially since the Reapers appeared to hide a lot of what it could really do.
But if you can't accept that, then let's look at Transwarp conduits from Star Trek, which weren't discovered until the Enterprise literally fell into one while in pursuit of a Borg Cube.
And they would be able to do this in ... three years? Seriously?
Just because you didn't read all the research papers discussing alternatives to Eezo, or see the work done doesn't mean they don't exist. For all you know people have been toying with the idea for over a thousand years, but it never got off the ground due to the prevalence of Eezo tech. The Reapers give people a reason to invest and they come up with a workable system in short order. I mean they converted weapons to Geth spec in less than two years so this won't be an issue.
Why haven't other cycles researched different methods? Having Eezo doesn't mean they don't research other things. Why don't we have a workable alternative?
Because the Reapers killed them before they got to far, I mean they did leave a guard dog to watch for that kind of thing.
Why haven't folks from Andromeda made the journey over here, if it's so feasible?
Because they don't have a compelling reason to leave...
As for using eezo in new ways, well, how many millions of years more advanced are the Reapers? And even they don't show any sign of intergalactic travel.
1. Programming error that made them focus on the Milkway.
2. They assumed they'd reached the pinnacle of development and thus saw no reason to move beyond what they could do. I mean it's stated that they spend the time between cycles powered down and conserving power. Doesn't exactly sound as if they're engaged in research and development
3. Trapped by their own purpose the Reapers failed to improve themselves, so it's not hard to see organics moving past their abilities with sufficient motivation. I mean we've seen how lazy and disinterested their creators were in doing anything themselves, not hard to see this trait transferring over.