The choosing of the soul is up to the people Upstairs or Downstairs. Higher and lesser beings that are in the realms that the writers don't like to write about unless their stories become embroiled in religious discussion that would ruin the fun factor. The Reapers' job is pick up and delivery, protection from the occasional evil spirit. There may be the occasional judgment call to allow someone to stay alive or do something pretty awful to those who had something to do with their demise.
I suppose with the many ways Death is Cheap in Fiction-land, Reapers can occasionally get either a mission to just keep company of the guy until he's brought back (maybe the occasional message), the mission is aborted (Reaper still gets full benefits of the job) or the Reaper is asked to deliver and once the revival happens the spirit is sent back.
There's still plenty of people in the galaxy to empower necro-gear and magic, the ones who "fall in the cracks", so to speak. The Reapers do get the occasional mission to frag Necro-users, though.
OK, maybe that rule can still be discussed. But there's something about having an adventure idea where, say, Bleach and Soul Eater Reapers go to war over some random schmuck that got run over by a van because they are falling behind in quotas that doesn't quite feels right with me. OK, sure, it's kind of hilarious in a way that reminds me of "Brazil" or maybe even "Paranoia", but also doesn't quite feels right.
Going back to "Bleach" for a sec, once upon a time I had this idea of "Bleach", you know, back when it was actually something worth saying you watched (and jumping too late on that salt wagon, if you ask me) where it is manifesting on the Line and there is a divide -- the old guard that is Soul Society exactly as you see it on the series and the new guard which is embracing stuff like Bankais manifesting as firearms and dressing in ways different from samurai (although there is still some military symbolism here and there). Both sides have heroes and villains, but the struggle is gonna go for a long time.