Plywood adds to strength but not too much, it does add to rigidity and helps keep thing square. If not supported underneath plywood will warp especially with weight on it.
The frame of the aquarium needs to be supported both horizontally and vertically directly under the frame of the aquarium... If the front of the tank is not supported by the frame of the stand it will rest on the cross members making the screws carry the weight. I know it is not easy to describe what I am saying.
In this picture you can see this stand is designed to stick out farther than the tank. They do this to have a stand larger than the tanks for more room inside the stand. The same thing could happen on the sides but just extend the red studs farther.
If you want just a overhang the bottom yellow studs could stop at verticals.
See my stand is actually two stand in one so I have two frames, but both stands share the same back.
Also everyone of these stands is pretty much over kill. My retail stand I just sold was made so cheaply. You probably could get away with allot.