Surely having one or more texts isn't a requirement for calling your beliefs a religion? To quote almighty Wikipedia: "[A] Religion is a collection of belief systems, cultural systems, and worldviews that relate humanity to spirituality and, sometimes, to moral values". I can't say I've ever heard anyone, including people studying religions, claim that the absence of such means it isn't a proper religion. If you are right, then neither the Native Americans, the Australian Aborigines, many of the pre-Christian Indo-European beliefs, nor today's remnants of primitive animism (to name just a few) can be said to be religions - which I really don't think is correct.
As for the purposes you outline for Ancient Greek religion/myth....it's pretty much the purpose religion still serves today.
Mythology, by the way, is a term only every used in hindsight, long after it has become obvious that the myths concerned are untrue. The average Greek, Roman, Persian, Egyptian, etc, etc certainly didn't consider their myths mythical, but to be true stories. Much, or rather exactly, like religious people of today consider their myths real.