Totally up to you whether or not to leave them!
I do have a Shadblow I have managed to train as a tree but it has taken me five years of judicious pruning to do it.

Thanks cearhaill. This is an Alnifolia Serviceberry. Previously I had pruned these new starts but the shrub has never really grew much. Recently I had someone tell me that by pruning the new shoots I was hindering its growth by not allowing it to grow as it wanted. Is that true?

Step away from the plant. I think you turned your Amelanchier alnifolia into a stolonifera. Your pruning could spur suckering, so give it a couple of years before applying anymore shears.

tj

Thanks for the advice! I will take a "wait and see" approach.