Assuming the string is also oak, I've had more success (in furniture making) when using a greater offset of something like 3mm. It sounds extreme but, in my experience, if you don't offset the holes enough, they won't have much of an effect at all. I'd try to make my own pins from straight-grained oak, instead of buying the cheap mass-produced beech dowels (...which you probably wouldn't be using on an oak staircase anyway! ;-)). You can very slightly taper of chamfer the ends of each dowel to help them locate in the tenon's holes and then, through in to the other side of the mortise. You're probably looking for 8mm dowels, maybe 10-12mm maximum.
As I said, though; most of my experience comes from making furniture on a relatively small scale. With large-scale joinery, things may be a little different...