My car has keyless entry but does not have the push button to start. You still need to put in the physical key and turn the ignition. My car has been broken into but without damage. They rummaged through stuff and took some random things but we keep nothing of value in the car. I'm not sure how they broke in but I've seen videos[1] online on how a tow service can get your car door open using an air-wedge. Maybe they did that, maybe they used something that repeated the key fob signal, not sure. But I'm glad that my car still needs a physical key. I'm not looking forward to the day when I need to get a new car and all that is available is keyless start. I'd happily go back to needing a key for everything, even the doors.
There is very little distinction between your physical key and a pushbutton. When you turn the key, it's just pushing a button internally that does the same as a button you'd hit with your finger. Few cars these days have any kind of direct connection between the ignition key and the starter.
Maybe I'm misunderstanding how the push button works because I've never owned a car that had one, but I thought the whole thing was that the key fob just needs to be near the car in order for the button to actually do anything. And these car thieves are repeating signals from key fobs that are inside the owners' homes. At least with a physical key, it needs to be present and inserted into the ignition to start the car, you can't use any technology to help there. Of course there are other ways to steal a car, but it at least deters these modern techniques.
1. https://www.youtube.com/watch?v=nEMzTDiXC6A