The significance of 334 BC is that it comes after the reestablishment of the shrine at Didyma, under Alexander, following its sacking by the Persians in about 494. The new system was not in continuity with the old. The prevailing opinion seems to be that the new shrine was operated similarly to Delphi, possibly including a woman as the main oracle figure, with male attendants. To the extent that there is a wider story of transition from male to female oracles, it is the story of the growing influence of Delphi specifically.
Literary and archaeological sources for the chronology are discussed in "The Branchidae at Didyma and in Sogdiana" by N. G. L. Hammond (Classical Quarterly 48(2):339-344, 1998). As the title of his article suggests, we believe that the priesthood or officials of the "old" shrine came from the family of the Branchidae, who after the capture of Miletus were transported by Xerxes to Sogdiana. There were many other hereditary priestly offices, so it is plausible that this shrine was also run on the basis of lineage, and that the loss of the lineage meant that oracular ritual could not continue. Herodotus is a major source for these events, but not the only one. Archaeologically, there is a corpus of a few dozen inscriptions from the 6th century BC to the 3rd AD, mainly recording questions and answers, which can be compared to similar material from other locations.
The subsequent lack of mention of the Branchidae in relation to the shrine suggests that in its revived form, they were not involved. But we do not have detailed knowledge of its procedures at any date, so it is not certain that the prophet of the old shrine was always male, and the new one always female. As summarized by Antti Lampinen ("Θεῷ μεμελημένε Φοίβῳ: Oracular Functionaries at Claros and Didyma in the Imperial Period", in Studies in Ancient Oracle and Divination, ed. Mika Kajava, Institutum Romanum Finlandiae, 2013; internal citations omitted),
It may be that he female prophetess of Didyma stems from the renovation of the oracle in the 330s, and probably had no precedent during the Branchidae tenure of the sanctuary – though technically such an assertion cannot be decisively made due to our lacunose sources. The cultic myth of the Branchidae, however, seems to imply a male seer in the oldest tradition of the sanctuary. The method of choosing the prophetess is still unknown, but it could hardly always have happened through divine intervention as in the case of Tryphosa, which is presented as a special circumstance, unless the formulation of the inscription (ἣν ὁ θεός χρησσμῶι κατέστησε) is simply a conventional form of referring to an election by lot. Some further influences from the Delphic establishment would not, however, be surprising. With regard to Delphi, for instance, there is clear evidence for several contemporary Pythias in the period of the sanctuary's greatest efflorescence, but so far our meagre evidence on the Didymaean prophetesses cannot be used for any trustworthy analogies.
The Tryphosa inscription mentioned is the same one being cited by Fontenrose in the question: a fragmentary text which nonetheless is clear that there was at least one female prophet. It is translated in Ancient Prophecy: Near Eastern, Biblical, and Greek Perspectives by Martti Nissinen (OUP, 2017) as:
Hydrophor of Artemis Pythie, Platainis Melas’ daughter, called Tryphosa, whose grandmother is the prophetess (prophētis) Tryphosa, whom the god appointed in an oracle, when Claudius Charmes the younger was prophētēs
As Lampinen discusses in further detail in his paper, there are many reasons to believe that the reestablished shrine was influenced by practice at Delphi - and that would comport with having a woman as the main seer. The violent end to the old shrine would give an opportunity for a break from whatever the practice used to be beforehand. However, since many of our sources are much later, it's also possible that their accounts are being shaped by the example of Delphi: tending to make Didyma sound more Delphic than it really was.
Either way, it seems like we have at least two interpretations -
- The refounded shrine was influenced by practice at Delphi, including a female oracle.
- It wasn't all that Delphic, but our later commentators viewed it through Delphi-tinted lenses.
which either way point to the general importance of Delphi.