Will the highlighted legs bow?
The legs will not bow at all, all the force acting on it is pushing straight down. There is nothing going on the inside to act upon it to push out, so this will hold anything you wish to place in it. It will also hold an incredible amount of weight on top without no bowing of the legs either, if you added an apron that is attached to the top the same way you have a front stretcher at the bottom. If you did the top a little differently and made it so the top blocks "keyed in, or just added a few removable screws, it would help hold the front together.That way you could still remove it when you need. Like this without the screws at the top, it will not bow, maybe spread a little if you don't fasten the top. 



I added your modified picture to help show what would help