The Capitol's decision to allow two victors in the 74th Hunger Games was motivated by the rebellion brewing in District 13. This rebellion threatened the Capitol's power and control, and the Capitol was desperate to quell it.
The Games' rule change was seen as a desperate attempt to appease the districts and prevent a full-blown uprising. It was not a result of Haymitch's influence, but rather a strategic move by the Capitol to maintain their grip on power.
Haymitch, while a skilled mentor, was not in a position to influence the Capitol's decisions to this extent. His primary role was to ensure the survival of his tributes, not to change the rules of the Games.