Spine is only how far that arrow will flex with a weight suspended below it. So a consistent spine is important and high quality arrows will have a more consistent spine.
However all arrows have a dynamic spine. This means when put under load an arrow wants to flex a certain way. If you find the sweat spot that allows all your arrows to flex the same way out of your bow it will tighten your groups.
BP is correct in that quality arrows will improve your groups anyways but you can still tighten the quality arrows up by doing what I suggested above.
It would be cool to see if when you do your spine test and find the stiff side and mark it some how then do the bare shaft test to see if the mark you put on the arrow always comes out either up or down in our out. Then you will know you can save some time shooting all your arrows through paper.. ive just never done it that way.
Also if you shoot a few 3d shoots and have someone impact your arrows you will notice you can you get a flier if that arrow is numbered. I cant tell you how often I have found a flier arrow and have either had to retune it or just throw it out. This is one of the major reasons I have practice arrows and tournament arrows. I also never shoot at the same dot. I just don't want to waste a lot of time tuning my arrows to my bow to have someone or myself impact my arrow rendering it suspect so to speak.
I have also seen guys try to do what I do to my arrows and it never works for them. I think it is because they are not consistent enough between shots. The hooter shooter is a great machine for tuning arrows. If you do some research on the spot hog web page you might find the article on tuning your arrows.
The harder you work the luckier you get!!