Did the invention of crowns/diadems come from a desire to ward off headaches?

Tiara

A Tiara is already mentioned by Herodotus, about 300 years earlier:

TIARA To the Greeks the τιάρα or τιάρας was known only as the head-dress of the Persians. Herodotus, whose information on this point, unlile that of most classical writers, is at first hand, says that it was of soft felt (7.61, τιάρας καλεομένους πίλους ἀπαγέας: cf. 3.12), and was worn by the Persians not only when campaigning, but at the more peaceful occupation of sacrificing (1.132: cf. Serv. ad Aen. 7.247).

However, this item takes more the shape of a helmet- or cap-like item, yet it already was made from gold as a status item on occasion, as Herodot depicts:

The men who served in the army were the following: the Persians were equipped in this way: they wore on their heads loose caps called tiaras, [...] - Hdt 5.61

There is further proof of this, for it is known that when Xerxes came to Abdera in his return, he made a compact of friendship with its people and gave them a golden sword and a gilt tiara. - Hdt 8.120

His nigh contemporary Xenophon remarks something, that makes it interpretable that the way to wear a tiara depends on the rank:

the King alone may wear upright the tiara that is upon the head, but another, too, with your help, might easily so wear the one that is upon the heart. - Xen. Anab. 2.5

Diadem (διάδημα)

On the other hand, the διάδημα/diadem can be found also 300 years before Diodorus in Euripides Medea:

I shall send them bearing gifts, [785] [bearing them to the bride so as not to be exiled,] a finely-woven gown and a diadem of beaten gold. If she takes this finery and puts it on, she will die a painful death, and likewise anyone who touches her: with such poisons will I smear these gifts. - Eur. Med. 764

Or even further back: Homer, around 800 BC, mentions a specific item that is translated as diadem in both the Illiad and Odyssee, without using the actual word:

Thereon Hera left them and resumed her seat inside, while Iris and Apollo made all haste on their way. When they reached many-fountained Ida, mother of wild beasts, they found Zeus seated on topmost Gargaros with a fragrant cloud encircling his head as with a diadem. They stood before his presence, and he was pleased with them for having been so quick in obeying the orders his wife had given them. - Hom. Il. 15.113

For I would, O father Zeus, and Athene, and Apollo, that no man of the Trojans might escape death, of all that there are, neither any of the Argives, but that we twain might escape destruction, [100] that alone we might loose the sacred diadem of Troy.” - Hom. Il. 16.74

But come, weave some plan by which I may requite them; and stand thyself by my side, and endue me with dauntless courage, even as when we loosed the bright diadem of Troy. - Hom. Od. 13.366

And Odysseus answered, "In good truth, goddess, it seems I should have come to much the same bad end in my own house as Agamemnon did, if you had not given me such timely information. Advise me how I shall best avenge myself. Stand by my side and put your courage into my heart as on the day when we loosed Troy's fair diadem from her brow. Help me now as you did then, and I will fight three hundred men, if you, goddess, will be with me." - Hom. Od. 13.9

A hundred years before Diodorus, Polibius did write his histories, actually using the word διάδημα:

He assumed the diadem, adopted the title of king, and was at this time the most powerful and formidable of all the kings and princes this side Taurus. - Plb. 4.48

As for Diodorus, he uses it without that indication of source later in his histories:

Alexander founded other cities also at the distance of a day's march from Alexandria. Here he settled seven thousand natives, three thousand of the camp followers, and volunteers from among the mercenaries.5 3 Then he marched his forces into Bactria, since news came that Bessus had assumed the diadem and was enrolling an army. - Diod. 17.83

Research history on the topic

Hans-Werner Ritter wrote Diadem und Königsherrschaft, a rather long German book on the topic of the diadem as insignia. He points to Xenophon:

Zum ersten mal findet sich das Wort διάδημα bei Xenophon in der Kyrupaideia(8,3,13) in einer Beschreibung des Ornates, den Kyros nach der Unterwerfung der Meder angelegt hatte(s.u.S.14 mit Anm.3). Er trug, heißt es da, eine aufrechte Tiara [...], wie ihn sonst niemand tragen durfte [...] „Er trug auch eine Binde um die Tiara [...] - p.6

A rough translation of that section:

The first time the word diadem in Xenophon is in the Kyrupaideia 8,3,13 in a description of the regalia [Ger:Ornat] that Kyros wore after capturing the Medeans. He wore an upright tiara [...] not allowed to others. "he also wore a band around the tiara [...]

The book itself is worth a read, if you can handle the constant Greek and are fluent in German. He alleges, that it is common consensus among historians, that the Diadem is of Persian origin¹, and that it can't be of old Macedonian origin²

Conclusion

Based on the way earlier use of the diadem before Diodorus as a regal item with both Xenophon and Polibius, I put that paragraph from Diodorus book 4 into the same category of historical fiction that we see all too often with Herodotus: an invention to capture the listener's attention. Ritter's work points to it being not of such medical origin, but a type of Persian regalia.

1 - Ritter, Hans-Werner: Diadem und Königsherrschaft, Pp. xiv + 191, Munich (1965), p.31.
2 - Ritter, Hans-Werner: Diadem und Königsherrschaft, Pp. xiv + 191, Munich (1965), p.34.