Actually I don't think that elevator pitch writes itself. It's not an inherently bad place to start but on its own it's pretty neutral. All the
amazing stuff you mentioned could easily be done without leaving the galaxy. As you point out, the game never earns the fact that it is taking place in another galaxy. From the motivations to go there to what it's like when you're there, it never makes for a compelling story. It's like at every turn when the designers had a choice, they went with the most boring option.
One thing I want to know is why go to Andromeda in the first place. Surely not all the star systems in the Milky Way have been explored. I might be wrong on this, but haven't only the planets near the Mass Effect relay network been explored? Doesn't that leave most of the galaxy a mystery? Why is exploring an unexplored planet in another galaxy more desirable than exploring an unexplored planet in our own galaxy? To launch such an expedition, something really unusual and important must have been detected on long range scans. Saying what that was and why it was important enough to launch such an expedition would be a movie elevator pitch that would get me to watch.
Before the reapers
Before the Citadel
Humanity discovered the first mass relay orbiting Pluto
For 3.47 minutes before power was connected during startup sequence
The impossible happened
A gate formed
Through it fell a dead half crystallized man
On his body was a map of a star cluster in the Andromeda galaxy
His name was Alec Ryder