I think you got two mixed up here. Aqeeqa or slaughtering of sheep was a practice even from pre-Islamic days among Arabs whenever a child was born in a house. It's a form of celebration. Aqeeqa means celebration. This was continued even after Islam came.
The hair is removed because that is the only sure and short way of cleaning pregnancy (pre-birth) impurities in the child. The weighing of hair and giving in charity silver equal to the weight of it is a form of Sadaqa.
Please note that none of the above are Wajib or Fard (obligatory), but removing hair is highly recommended as soon as possible, to clean the child because Najasat can invite Shaitan. All the above need not be done at the same time.