That bathroom shelf stand will go sideways in seconds without a basic securing for lateral movement. The tension rod idea shelf would be much more solid using a support lathe secured with alien tape to the side of the washer and screwed into the wall on the other side.
The real question is, how long have these lasted? These are not as sturdy it have any real support, they are just for looks. If you want to make them more useful, you have to use much more support.
So instead of the tension rod idea you propose the cart at the end which is worlds better but why not make the shelf's adjustable and "captive" by using pilaster at the front and back and vertical braces which would add rigidity to it
