When I make this "toothpick", is there any need to make it so the grain matches? I can't imagine a toothpick with grain running along its axis and thus being cross grain to the stock doing anything but being very noticeable when stained.
Yes, generally you should try to match the grain. When I rough out a stock, I save large pieces prom the various areas of the stock and mark them for orientation. That way, if I need to make a patch, I have a scrap from the same general area that I can line up so the grain matches.
However, from your picture it looks like the patch will be in a stripe which will take stain darker anyway, so end grain might work just as well.
Dale H