Although I don't use any electronic aids during my sessions I could see an application where each round every player clicks his modified iniative roll and then the dm gets his iniative lists.
Yeah, I could almost see it working if you had an electronic aid, where at the beginning of combat the DM has everyone's default initiative modifier and weapon speed inputed into an app, and instead of every player rolling each round the DM simply announces the next character's turn in sequence after the app rolls for everyone at once. If a player announces a quicker action at the beginning of the round the dm simply clicks a modifier for that round. But if everyone insists on rolling his or her own initiative (as most veteran players would probably expect) it just starts taking up more and more time. Also, you'd probably want a timer for everyone to plan their actions for that turn, lest it get drawn out interminably.
Pretty niche though.