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It would take a literature search to prove it, but I think that the claimed etymology is not precisely correct for English. It is often the case for English that a word is adopted and then a mutation of it happens independently of mutations in the original language.

As an alternative source for arrogant, please consider arrogate:

 > https://www.merriam-webster.com
 > arrogate: [verb] to claim or seize without justification. to make undue claims to having : assume.

I think Ayto is skipping arrogate as the source of the modern English arrogant because it might look like an unnecessary waypoint. However, the semantic similarity is important: arrogant is an adjectival form of the verb arrogate. An -ant ending signifies adjectives concerned with states of being: immigrant, contaminant, dominant.

The ad- prefix means "toward", specifically, rather than the indicator of an infinitive one might otherwise assume in English.