Wow! There's a lot of wood in that table, I imagine its quite substantial and heavy. It looks solid!

Now for the the bad part <well, maybe not bad>.... that's an unfinished table, so that's why you are getting the tiny splints of wood when you wipe down the surface. As you continue to use the table, its going to soak up every small stain and dropping on the top surface, and any mopping near the base will get a water soak as well. This is OK if you want an "Old School' tavern look,but be prepared for a hundred glass rings on the top and butter-grease marks, etc.

Solutions? Not many and keep the table as it is.

1) You could put a glass top on it, edge to edge.

2) Though it will change the look, feel and may darken it some, I would consider applying a coat of paste varnish on the entire piece. Use a can of clear (paste only) varnish and get a dry rag. Apply ONLY with the dry rag and very sparingly. Do not 'coat' the piece with the paste varish, rather dry-rub it onto the table in the direction of the grain. You want to apply it DRY-DRY-DRY. Just barely enough to cover. This will lock down the loose fibers on the top and also give you some stain resistance. However, before you do this be sure you do not want to return it, and start out on the underside of the table to see what it does to the color and if you like the look when it dries.

All based on the photos you sent, but that's my read on it.