There are a couple of situations where you should always consider using knotting. One is on new wood and the other is where the wood is going to be exposed to sunlight.
Since your staircase doesn’t isn’t affected by either of these issues you are probably safe to carry on without it. There is always a slight chance you’ll get a bit of discoloration over time, particularly with white paint, but it’s unlikely to be severe or particularly noticeable.