A few reasons:
1. It keeps whatever happened in the Milky Way as still having happened. While this is a negative to you, it is a positive to many for reasons that have nothing to do with an ending. There's a reason so many people dislike the constant reboots and AUs of comics, mostly because they feel it devalues the universe by not having one unified timeline.
2. It allows you to construct new ideas for races, civilization hubs, conflicts, etc. A reboot that still takes place in the Milky Way makes sense if you keep things like the Citadel and Mass Relays, but at that point it's too much of the same. A new galaxy allows you to try new things and establish new histories, since presumably it would take place enough in the future that any relocation of Milky Way residents to the new galaxy will have resulted in an up-and-running sense of civilization.
But in the end, I have to ask you, why shouldn't they relocate? It seems to me the core of your entire preference is to erase what happened in the trilogy so you don't have to think about what happened during the Reaper war. I don't find that to be particularly compelling for why they shouldn't do a new galaxy instead.
On my end, the reason why I prefer new galaxy is because I hate alternate timelines. I despise them, and I'll never read an X-Men comic because of them.
1) True. But at the same time, it's very limiting for the writers. They can't do anything with real consequences as long as everyone has to start out at th esame point for the next game.
2) You can do the exact same thing in the Milky Way. Only 1% of the galaxy had been explored by ME3, and we barely visited a fraction of that. There's plenty of space fro new civilizations, alien species, worlds, etc. And keep the relays, since that's a staple of the Mass Effect series
Galaxies are freaking BIG
Reaper tech's been reverse-engineered ever since the cycle got its hand on Sovereign's corpse. It's frankly less BS of an excuse than "ohh this region of Milky Way just so happens to be unaffected by Crucible wave despite the whole galaxy being covered in it because reasons".
Well, except the Reapers explicitly violate known physics by doing what they do:
Reaper power sources seem to violate known physical laws. Reapers usually destroy fuel infrastructure rather than attempting to capture it intact, indicating that Reapers do not require organic species' energy supplies. Consequently, the Reapers attack without regard for maintaining supply lines behind them, except to move husks from one planet to another. Unlike Citadel ships, Reapers do not appear to discharge static buildup from their drive cores, although they sometimes appear wreathed in static discharge when they land on planets.
Looks like organics would fry if they tried to do go too long without a discharge, even with Reaper tech.