I find the assumption this weaving scene represents the competition of Athena and Arachne odd, to say the least, so I did some research.
Here is the description of Object CP 2038 by Gladys Davidson Weinberg and Saul S. Weinberg (1956):
See also: Williams, Charles Kaufman, J. Lawrence Angel, Peter Burns, and Joan E. Fisher. “Corinth, 1972: The Forum Area.” Hesperia: The Journal of the American School of Classical Studies at Athens 42, no. 1 (1973): p. 13. https://doi.org/10.2307/147469.
Here's the description of the same vase from ASCSA: (https://ascsa.net/id/corinth/object/cp%202038)
woman to rt., weaving at vertical loom; long hair to below waist, wearing chiton and necklace; holds 2 rods. At rt. 2nd woman facing lt., standing on stool (?) with shorter hair, unbelted garment, holding 2 forked tools against loom. Loom: 2 uprights and horizontal cross-bar, loomweights holding warp threads, length of woven cloth at top with incised cross-hatching. Surface damaged to rt. of her: 3rd woman facing lt., possibly holding loomweights; 4th woman, very tall facing rt.; small 5th woman facing 4th; behind, 2nd loom like 1st.
As OP notes, there's nothing in this scene to relate it to the contest between Arachne and Athena. There are two looms and five women weaving, one of whom is apparently a child who has to stand on a stool to reach the weft.
Also, an aryballos is a container for perfume. It's hard to understand why such an object would be decorated with a cautionary tale, but makes better sense if the decoration is a domestic scene of women of various ages performing the normal household task of weaving.
I think that the Weinberg's interpretation of this scene is inaccurate.
I checked a few websites (the perseus.tufts.edu Art & Archaeology Artifact Browser, The Beazley Archive Pottery Database, the British Museum, and Wikimedia Commons - okay, not exhaustive, but still) and the only ancient depiction of the contest I could find is a frieze of Arachne and Minerva from the Forum of Nerva (built 85 - 97 C.E):
"Rome, Forum of Nerva, so-called "Colonnacce", detail of the frieze, scene with Arachne's punishment." Image source: https://commons.wikimedia.org/wiki/File:RomaForoNervaColonnacceFregio2cAracne.jpg Image license: Creative Commons Attribution-Share Alike 4.0 International
Interesting commentary on this section of the frieze and Ovid's myth here: https://penelope.uchicago.edu/~grout/encyclopaedia_romana/imperialfora/nerva/colonnacce.html
Good investigative work, OP!
If it's any consolation, I emailed a museum to ask about a particular object several years ago (it had been repatriated from the Getty and I was hoping they a photo) and never heard back from them, either.