Dowel not only doesn't match the grain, but it is weaker when screwed into. Drive a screw into the end of a 2x4 and one into the side, then take your claw hammer and pull them out. The end grain screw will come out much easier. Don't take my word for it, try it. Why do people go to great lengths to dovetail a box together, put step joints on them and all the other work? Because screwing into end grain wood is not as strong.
When you're plugging to re-screw, the plug might need to be deeper than the plugs you can buy. For example, if the old screw was 5/8" into the wood you want to drill your plug hole to the bottom of the screw hole, 5/8" in this case. The bag of plugs you buy will probably not be longer than 3/8". So you'll want to double plug. Carefully flatten both ends of the first plug by sanding as needed, use a little epoxy on it and slide it in. Then take the second plug, with at least one flat end, epoxy it and slide it in to make contact with the first plug and tap gently to seat them both in the hole. Clean up the overflowed epoxy, stain and finish as needed.
If you have an ovaled hole, maybe the screw backed out and bent over or whatever, instead of using one large plug use two or more plugs, "stepping" them. This also works with smaller non structural cracks. Start at the end and make your first hole, plug as needed. Then move over 2/3 the diameter of the plug and make your next hole. This hole will cut away part of the 1st plug. Epoxy plug #2 in and continue stepping as many plugs as you need. If you're repairing a fairly long crack in a board you don't want to replace, you can even make patterns. Start at both ends and work towards a final plug in the middle. Use two different sized plugs, alternating, etc.
Another tip. If you are not refinishing the entire piece, just repairing a hole here or there, before you drill put a wide piece of masking tape over the bad screw hole, then do your drilling thru the tape. Now you have a perfectly masked hole to clean the excess epoxy off. This makes your spot refinishing much easier.
If you epoxy the plug into the hole (drilled to correct size for the plug of course) the epoxy bond is stronger than the wood, so it's not going to give. The grain running sideways, as is the grain in the rest of the piece you're repairing, will be as strong as it originally was, and look, depending on the plug and the skill that it was installed with, almost as if there was no repair.