Rented a Nissan Pathfinder (NOT any of the names below-but had to click on one) recently, and battery went dead. When AAA guy came to jumpstart, he said batteries can go dead on pushbutton ignition cars, if you don’t put your foot on the brake when you turn it off. Never heard that - but can’t think of another reason that battery died, which created a big hassle with Hertz, who tried to make it my fault. (And it wasn’t - lights were out, doors were locked, all was well - except car was dead in the AM.) Your POV?
You don’t have to apply the brake when shutting off the vehicle but if you push the button more than once you can switch the ignition back on and that will discharge the battery if you leave it on and walk away. If you apply the brake while switching off the vehicle and push the button more than once you may hear the engine start, you will notice that the ignition is on in that case.
You may never know if the battery was discharged or if it failed but why would Hertz care either way?