Most plywood just won't engrave very well, as you've noticed. The cheaper grades are especially bad in this regard. If you need to use plywood in this project, find one with a fine-grained hardwood top veneer. In the US, that would be cherry or maple. In Greece, maybe Baltic Birch or something else - you might have to experiment to find something that works. Get the best and sharpest V-carving bit you can find. It doesn't have to be carbide; usually a HSS or cobalt steel bit will be sharper (although it won't last as long). But it's unlikely to be Chinese.