Other than Prabhupada's ISKCON commentary, pretty much all other commentators on the Bhagavad Gita interpret these verses as not referring to the time of death, but rather to the names of the series of gods who escort the soul on the path to Brahmaloka/Moksha. This path is described in the Panchagni Vidya of the Chandogya and Brihadaranyaka Upanishads, as I discuss in this questionthis question, this questionthis question, and this questionthis question. (I also posted this questionthis question about the origin of the Panchagni Vidya.)
Note that the terms in this passage, like "day" and "year" aren't destinations as such; it's not like there's a place called "the day" that the soul goes to. Rather, it refers to the god of the day, the god of the year, the god of lightning etc. These gods are involved in escorting the soul to Brahmaloka/Moksha: the first god leads the soul to a certain place, then the second god leads it even further, etc., until finally a being whom I discuss herehere escorts it to the final destination. See Adi Shankaracharya's commentary on the Brahma Sutras here and here.
By the way, it should be noted that while Ramanujacharya and Madhvacharya see this path as the direct path to Moksha, Adi Shankaracharya only sees this path as a path to Brahmaloka, as I discuss herehere and herehere. Now Adi Shankaracharya did believe that the inhabitants of Brahmaloka would eventually get Moksha once Brahma dies at the end of the Mahakalpa, but it's a much slower route than just getting Moksha as soon as you die. So insofar as Vivekananda and the other people you mentioned were Advaitins who wanted Moksha as soon as they died, the Gita verses we're discussing would be irrelevant to their goals.