On a mass produced bottom rimmed tank (where the bottom panel is captured between the sides vs. being underneath it) supporting "just the corners" is pretty much OK. It just doesn't leave much to resist racking, and it doesn't make sense (to me anyways) to design in a foolish manner like that. I'm guessing the tank's bottom rim rests along the lip inside each end panel, and along the (very short) tops of the two vertical posts on the back? And also the top of the brace in the middle at the front only? It doesn't touch anything else besides those points?
If I were in your shoes, I'd laminate a piece of 3/4" ply along the back, connecting the two narrow verticals and providing a constant edge along the back of the tank for support. I'd make it 8" tall to give stiffness and a long joint to the verticals in each corner. I'd remove the center brace and add a narrow shim along the top of the inner front rail whatever thickness was required to give contact to the full front rim on the tank. Then, replace the center brace with a piece of similar dimension, but oriented vertically and screwed to the front and back rails. This will keep them from going anywhere.
These changes should be quick and easy and at that point, the stand would have full contact around the perimeter of the tank, and it would be better braced for racking thanks to the wider support along the back. In terms of being able to carry the vertical load, I don't think you've got any cause for concern or need to beef up the corners at all.
