I'm not sure which direct connection you're thinking of, but for most phones that would be limited to WiFi (probably WiFi Direct), Bluetooth, and maybe NFC. NFC range is tiny and Bluetooth's is pretty small. WiFi's range is approximately the same thing as an RF remote, which isn't great.
Also, if we did have direct connection (which would be great for confirming the start worked, and the status of the car), why would we need internet??