Yes, it launches in full screen just as a video would. I just have to tap the button to start the game, it instantly gets maximized.
The way you describe it, I would also assume that it is either an OS or browser related issue. I sadly have no knowledge about Apple/Safari, beside the fact that Android and iOS apps are two different things to do. You would assume HTML5 gets rid of incompatibilities, but maybe it is not that easy. Maybe you can find something of note by doing a google research?