>I'm not sure what happens if one of your peers sends you a block while you are working through your outputs. Maybe you stop working and use the peer's block, or maybe you keep working hoping to find an output of your own which can compete with the peer's block.

This to me is the difference.  A single user may finish hashing their wallet in under a millisecond vs JD jumbo wallet which takes 3-4 secs.  A 4 second head start would allow a valid stake to completely update the entire network me thinks.  It might make sense for client continues to look for valid stakes for a few 100 milliseconds because the time to propagating a valid stake is kind of variable, but after that it seems like it would be a wasted effort.  I do not have the knowledge to read code to know how clam works.