Realistically, I think it boils down to the fact that a 3rd year will absolutely not hurt your chances. It is only incremental gain. However, your decision to apply after 2 or 3 should come down to whether or not you have enough quality experience (both in and out of work) after 2 years. I know plenty of people who have gone to business school 2 years out of finance/consulting, and all of them were very involved outside of work, and have been very successful. I think it's case-by-case, and you just have to see if you have enough experience outside of solely finance/consulting to justify applying "early". This is of course, not to say that people who wait 3 years are not qualified to apply after 2-- they are equally as talented, many have just chosen to work an additional year.
But speaking to the other poster's point, I think she's absolutely right in that there is increased responsibility in your 3rd year. It just depends if those talking points are pivotal to your application.
Additionally, I don't think you should get too hung up on "managing others". Realistically, adcoms will know very well that any analyst, no matter how senior-- is not going to be bringing a tremendous amount of managing experience to the table. There will be people in your classroom who have managed entire departments and divisions-- and that's an expertise that they will readily bring to the table. Informal mentorship and managing others is great, but this is also something that can be replicated through extracurricular activities and leadership positions.
Just my humble opinion.
Good luck!