'thrice-garlanded'
I fully agree with your line of thinking, Joshua.
The sentence ending with "ὠθεῖσθαι / to be pushed" has led most translators toward the "wheel" interpretation -- but it can just as easily mean "compelled."
With that problem out of the way, τρικύλιστος certainly has a basic meaning of "rolled three times" -- but with κυλιστός meaning "twined in a circle, an epithet of a kind of garland" (as LSJ states, and as you and Athenaîos have shown) -- "rolled" can certainly refer to a garland. (It seems most of the "garlands" Athenaîos is discussing are worn on the head, so I think we are talking about wreaths.)
So the only remaining mystery is the prefix "three" -- which could easily be a colloquialism for either entwining three different plants to make one wreath, or just placing one upon the other -- the effect would be similar.
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Regarding "Three," we do see our Philōnídēs of Laodíkeia honored with three distinct wreaths (στέφανοι) in the inscription Attica, IG II2 1236:
"...the people honored him: both [the elder] Philōnídēs himself and his sons Philōnídēs and Dicaearchus – with citizenship, and crowned them with (1) a wreath of laurel and again with (2) a golden wreath... ...to praise [the elder] Philōnídēs of Laodicea and his sons Philōnídēs and Dicaearchus, and to crown each of them with (3) a wreath of myrtle, which it is customary to use when crowning their own benefactors."
Significantly, these are specifically honors given to Philōnídēs by the Athenians.
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So it could mean, in the context of Epikouros' letter to Themísta, "with all the honors you have given me in your letter" (i.e., crowned with multiple compliments)"