No, This doesn't look like the guy glopped lacquer on. Besides, there are techniques of lacquer ware that call for hundreds of coats of paint which are then sanded to create interesting patterns. In the right panel, you can see that the material was a fir plywood. There isn't much you can do about it. Yes, you can do the painting and sanding, but as the wood swells and shrinks according to the humidity of the seasons, the grain pattern will return. Use a cabinet grade plywood such as a birch or maple plywood or even MDF which has no grain. There is almost no grain to these plywoods, and what grain there is can easily be filled and painted out.
If you have already started with fir plywood, you have a couple of choices: Start over or veneer your wood. The number of layers of paint you need to adequately smooth out the surface of fir plywood will cost comparable to a laminate surface. Formica style laminate comes in three grades: Vertical (thin), Horizontal (med) and countertop (thick). You can use the Vertical or "V" grade on most, if not all surfaces. Just be careful about getting an even glue surface because the thinness of the laminate will show unevenness of the adhesive. You can get laminate in any number of colors and patterns. a second type you can get is called backer board which is simply the laminate with the face sanded so that you can paint it. We use it a lot in the entertainment industry with great results.