This is something which I have never really liked. It's the whole "are you mochel me?" marathon that those fearful of the sensation of burning flesh try to rush in right before Judgement Day. You know they're not sincere because _they_ don't actually think they've done anything wrong and even if they do suspect that they've done something wrong, you know they're only asking you because they fear divine retribution, not because they're actually sorry.
What then does this whole race accomplish? It's of the same vein as the mumbling Ma'ariv and the fast without a cause. The mentality is "I did the duty, now I'm in the clear." If people really believe that God cares so much about their individual conduct, do they think He's fooled by such transparent reasonings? Now I'm feeling a little bit like Isaiah here, but ritual without content is pointless and is at worst offensive.
Asking people for forgiveness could be done the right way with actual intent and with actual care about how one's conduct affects another. Though I do prefer to let bygones be bygones than to resurface old scars between people. But still, it isn't even wrong that such thoughts of personal accounting could be brought about by a certain time of year. And in that sense, Yom Kippur can be a meaningful holiday dealing with individual introspection and measuring oneself up with how one has been and how one wants to be. We can improve ourselves and we can become better people.
Begging the bearer of a divine Sword of Damocles for mercy is how most of Orthodoxy sees Yom Kippur for your fate is sealed on that day, but again, this is not about improving yourself for morality's sake, but for placating an angry overlord to whom prayer is a meager substitute for the full bodied aroma of barbecued animals. I think this is a weak and wrong-headed method for improving behavior. Most people will quickly revert to their previously comfortable levels of moral observance. One really must be honest with oneself and think deeply about what they may have done and how to improve themselves. You can't ask for mercy from yourself, but you can promise yourself to do better.