What would you recommend for wood? I was thinking plywood and wrapping it.
Now the sides and back of your tank are being supported by the sides and back of the stand. However the tank is also pressing down in the middle of the stand, and the front edge. This is why many stands put a support beam (or more than one depending on the size of the tank) in the middle of of the stand, running beneath the top, to support that weight and keep the middle of the top of the stand from bowing.
When you need to put in a support beam to support the second floor of a house, you have to take into consideration the cross-sectional strength of the beam, the length of the beam, and the end supports of the beam that transfer the load down to your foundation. That is exactly what the front of this stand is - it is an open space being supported by a beam that runs along the front edge of the top. You have a 5" wide piece of laminate particleboard that is perhaps 1/2" thick, and who knows how it is affixed to the sides and the top, because I can guarantee you those joints are not designed to bear the weight of a 250 lb aquarium 24/7. You can imagine that even if it holds at the beginning, with time that trim piece may pull away from either side, or the top, and when it does, your entire aquarium is going to end up on the floor.
Now if you want to keep the clean lines of that stand while improving the structural integrity, you could just build an internal frame within the stand out of 2x4's. You would see it when you open the doors, but otherwise it would be invisible. Rather than screw the frame to the stand walls, use polyurethane adhesive - stronger and waterproof and you don't have to worry about unsightly screw holes.
There are a lot of ways you could add strength to the setup... but all of them are going to require a little space within the stand. So if you specced your sump so that the sump fits every square inch within the stand... you may be in trouble.