That's how most newer push-to-start cars are, not just Mazda,and it's both aftermarket and OEM remote start systems. As you noted, it's for security purposes because you can get proximity keys online.
I think the only push-to-starts that don't shut off when the door opens is Nissan. But even still, you have to do the takeover process within 30 seconds iirc.
turning your car on, right after it was shut off by opening the door, is probably no good for the engine, so you would have to wait 30 seconds to 1 minute to turn it on again, which defeats the purpose of a remote start. I just don't think is worth the price, with dealer installation and all, it would probably cost $-700800 just so you can get fresh more easily in hot weather or more warm in cold weather
I live in Florida, so I don't need it to warm me up, it may be helpful to cool down the car, but you can do that why opening the windows and running auto mode, sparring yourself from waiting a couple of minutes inside the car for it to cool down or get hot is not worth $800
cool to have if you have money to burn, rather use those $800 towards a full suspension setup or something more meaningful, or even use it for a 2 way alarm system