For me, I needed a saddle with center cutout in order to do invisible aerobars comfortably, and yes, it's less stable so I only do it on long, straight stretches of road. On the indoor trainer I've also tried wrapping extra padding where my forearms rest (e.g. a sock) and it adds to comfort.
But now I've gone full aerobars on both outdoor and indoor bike so I don't practice that position anymore. Though experience with that position does help with transition to aerobar handling.