The OED calls this usage a pronoun. Here's the whole taxonomy of the usage you cited, followed by one of the many examples there.
Entry: other, adj., pron., and n., and adv. 2
B. pron. and n.
I. As pronoun.
6. Another person; someone else; anyone else.
b. With plural sense.
(b) In form others (also genitive plural others', formerly others.) (The regular modern form.)
1962 E. Waugh Diary 5 Oct. (1976) 790 She can write, think and pray exclusively of others; dreams are all egocentric.
One reason we might posit for the analysis as a pronoun is that we are not thinking of an indefinite count noun "others", such that you could point to the specific others and count them and distinguish them from the particular "others" that someone else thinks of.1 Instead, there is only one "others" that we all share, and it means "The set of people excluding the speaker."2
It does also have a common noun sense with various meanings; one good example is this:
1988 P. Brown Body & Society (1989) ii. 39 Slaves were the second inferior other in the world of the free male.
Meanwhile, it's clear that in your sentence it isn't a determiner nor an adjective. One reason is that neither category is inflected in English, hence others is not a possible form of the adjective other.
Whether the noun and pronoun grew out of the determiner sense or vice versa, or whether some kind of ellipsis always occurs as you propose in your question, is a more academic question. The OED has many senses of both the pronoun and the adjective going back to early Old English.
1 Of course, there is also another sense "the others", as in "You're the only one home? Where are the others?" This does refer to a particular group of people who could be identified and counted.
2 Technically this set picks out different entities for every speaker,
but that doesn't make it any more a common noun than it does the phrase "everyone else".