I think it would be interesting to have a list of personal stats after a level or after a tier run or so that tells you things like how many monsters you killed or how many times you died from blundering into spikes. I realize there are lot of possibilities for tallies, but they can be level-specific e.g. Flame Lash Arena can have a tally of how many times you set yourself on fire. I like to laugh at my own failures so something like that would be entertaining :) Then there could be achievements like go through FSC without bursting into flames or go through Concrete Jungle without moving a single grim totem. I guess these would just be some little things that people can try to accomplish throughout their daily runs.
Nothing too crazy for overall stats but the ones that I could see being not so niche in terms of showing scores for:
I think a simpler tally of damage done to damage received.
Revives and deaths.
Number of monster types killed or just general number of monsters killed.
The only issue I have with this is competitiveness for kills/bragging rights might actually be detrimental to teamwork. A mad dash to kill things might jeopardize your team mates.
Other than that +1. It seems like it would be a fun mechanic. Though it has been touched upon already a few times.