Ancient Greek religion had no scripture, no congregation, no weekly meeting and no conversion. It was something a city did. That is the central problem for anyone reviving it, and it is not a problem of missing information.
Cult, not creed
Worship meant sacrifice at a specific altar to a specific local form of a god, performed by officials of the city on behalf of the city, on a calendar the city set. Athena Polias at Athens and Athena Alea at Tegea were not interchangeable. There was no orthodoxy to hold and no membership to join, and the question of what a Greek believed is largely the wrong question to ask of the evidence.
What a modern practitioner can rebuild
A great deal is recoverable. Calendars, festival programmes, hymns, prayer formulas, the correct handling of offerings, and the distinction between the Olympian and the chthonic rites are all documented in inscriptions and literature, and contemporary groups work from them carefully. What cannot be reproduced is the polis itself: the civic body whose officers performed the rites and for whom they were performed. Household and personal practice transfers; the public frame does not.
Sacrifice, and the substitution
The central act of the ancient cult was animal sacrifice, followed by a shared meal, and almost no modern group performs it. Most substitute libations, incense, grain and cakes, all of which are attested, and are explicit that this is an adaptation rather than the ancient practice. The candour is worth noting: a reconstruction that names its own substitutions is doing something different from one that quietly redefines what the ancients did.