I had toyota and aftermarket remote starters in my '13 Avalon, which had the push button ignition (fob in your pocket). Both systems, the car shut off when you open the door. Aftermarket had WAY better range though.
My TRDP has a key, not a fob, and I have the aftermarket starter installed by the dealer. One press of the starter fob (separate from key), and it unlocks the doors. Get in, key in ignition, foot on brake, and the remote start disengages, but truck keeps running, the way it should be.
Why they can't figure out how to make it work with the push button ignition is beyond me. You get in, hit the brake, and if it doesn't detect your key, it shuts off. Seems pretty simple, and my friend's MDX works like that.