Purely from a scientific standpoint, with no reference to or knowledge of feudal Japanese research, thinking or custom on this subject, here's what we know today.

If you have a solid piece and a fluted or "grooved" piece which are the same material, etc. then the solid piece is still stronger and stiffer.

However, it is also inevitably heavier.

The fluting or grooving, when done right, is simply a way of removing weight without losing nearly as much strength as you would lose by simply removing the same amount of material in all dimensions (scaling down.)

In other words, if done without carefully planned fluting, scaling a blade design down from 4 lbs to 3 lbs cannot help but significantly weaken the blade and make it more flexible as well.

If done by leaving the blade the same general size and shape but removing grooves or flutes from it, then some strength is still inevitably lost, but it should be a smaller amount. The geometric strength improves to make up for the loss of "mass strength" if there is such a term.

This is why many firearms have "Fluted Target Barrels" but most really serious accuracy competitors don't. The flutes allow you to save weight without losing much stiffness over a thick, heavy "bull barrel" but the plain thick barrel is still stiffer than the fluted one. So people who don't care about the weight of the weapon but need absolute accuracy are better served with the heavy barrel. People who don't want the looseness of a plain factory "pencil" barrel but also don't want to carry a 10-lb rifle all day as they stalk groundhogs, opt for fluted barrels.

Again, I don't know about the Japanese, but I've never heard it said that the Scandinavians, who were famous for fluted blades, used the flutes to hide imperfections. They were not nearly as concerned with matters of polish and finish as the Japanese seem to have been, but they saw value in the flutes to keep the weight of the swords low while not giving up too much strength or stiffness.