Thank you for making a bug report!
This is a scenario that would greatly benefit from a replay. You are correct, Magicians’ souls does send for cost and should allow you to send the Skill drain in order to draw cards as apart of its effect.
However, as there is no replay supplied, I can only speculate as to what the exact circumstances are. If you have a replay code for this duel, please supply it so that we can investigate and find out if there was something preventing you from drawing, or sending cards to the graveyard, such as Droll & Lock Bird or Dimensional Shifter or Macro Cosmos.
Please follow the provided format in the Bug report topic creation and supply replay codes when possible.
I will attempt to check to see if this is bugged or not, but at this time I do not believe it to be so.