Burrs were very commonly oiled to make the most of their grain so that's how I'd start. This is with a plain oil, mostly some type of linseed in the past in our part of the world, not a blended oil finish like the Osmo but I don't see any reason this wouldn't pop the grain just as well as a straight oil. It's mostly the matt finish that would lessen the effect, same as a matt finish makes any wood look a little flatter or duller.
You can continue oiling if you want a straight oil finish which can be quite durable if you build it up, and it's possible to achieve a pretty good gloss on harder woods, if you're willing to take the time to do it right. Figure it'll take at least a couple of weeks, longer if your workshop is cold and you have to wait longer between coats. I wouldn't bother myself, I'd oil to pop the grain and then switch to wiping on varnish immediately. This will give you the same gloss as many applications of oil in a fraction of the time (two or three days), with far less sweat expended and giving better resistance to scratches and water.
Although it is good practice to grain fill with an oil finish whether to fill grain is largely a matter of taste. On furniture pieces I don't care for voids showing, but obviously other people don't mind it. If you want the surface gloss uninterrupted by little dark specs then yes, fill.