How-Where should I end this partial skirt-board at the stair? (I have more wood to re-cut a new piece if need be)
I think this is a perfectly good solution as pictured. You could, I suppose, run a little longer at the yellow arrow, but I don't think you'd want to go all the way to a point. 

The only thing that offends this trim carpenter is the direction of the grain... I'd have done it in line with the angle of the stairs. (So no end grain on top.) 

But a bit of paint, coupled with the fact that this is the stairwell to the basement and anybody else looking at it will think it's marvelous... well, I'd say you're done.