Gap between dowel and the hole in a breadboard

I don’t know that variety of wood you are using, but here’s what works for me if it helps, using my customary British/European Oak where the final appearance is important. Sometimes a light finger-smear of natural oil on the dowel sides may help, but nothing that will discolour the wood. I’m thinking of Linseed oil or something similar.

1 – Harder dowel-wood, preferably the same species if you want to preserve the colouring. Alternatively a contrasting variety. Slower grown wood where the growth-rings are close together is often harder than adjacent wider, faster-grown fibres from the same tree.

2 – Chisel a longer, more shallow lead-in on the dowel end to avoid abrupt compression of the fibres where they contact the tenon hole. If there’s a tendency to damage the dowel sides, try relieving a chamfer onto the top corner of the hole in the tenon where it first comes into contact with the peg. This will avoid a crease mark as you knock the dowel home.

3 – Experiment with the gap-overlap on the tenon; you need less offset with harder woods. It only needs to slightly exceed the anticipated lateral, long-grain, shrinkage on the board end beside the mortise. The end-on tenon grain in the table won’t move.

The joint doesn’t need to be too tight. The object is to allow sideways movement as the table boards expand/contract seasonally. Remember, even after flattening and sanding, the dowel end will usually stand a little proud of the board after a couple of years. Don’t forget to elongate the tenon-holes near the edges or the table top will split. The middle hole needs to be an exact fit.

Good luck

No problem.
Unlike doors and structural joints, this arrangement of grain meeting at right-angles doesn’t need to be rammed up tight…. it needs to fit, but be able to move imperceptible amounts over the seasons.

Good luck