The method has a problem built into it and the tradition generally says so. The ancient cult ended, by most historical accounts, more than a millennium before the modern revival began, and no continuous line of practitioners carried it across the gap. What survives is fragmentary and heavily weighted towards Athens, towards the classical period and towards public civic religion, so household practice, regional variation and the ordinary religion of people who left no inscriptions are exactly what is missing. A reconstruction is therefore assembled out of holes as much as out of evidence, and every practitioner has to decide what to do where the record stops.
The answer usually given is that the attempt is itself the offering. Kharis, reciprocity between people and gods, is the framework: an offering is made, it is received, and the relationship continues. What matters on that account is that the honour is given in the form the gods were accustomed to receive it, as closely as can be established, rather than that a practitioner holds a particular belief. Orthopraxy comes before orthodoxy, and the ancient sources support that emphasis.
The same method is used well beyond Hellenic practice. Heathen and Asatru groups reconstruct from the Eddas, sagas and law codes; Religio Romana from the Roman calendar and antiquarian writers; Kemetic and Rodnover groups from their own bodies of evidence. Each faces the same argument about how to handle personal revelation where the sources are silent.