As I wrote in the comments my understanding always was that it was Jacob, who brought DHARMA to the island and this is how they fit the universe. In the end they were not different from the 'Oceanic 816' survivors or the Spanish ship crew. They were people that Jacob wanted on The Island. Whether it was him behind DHARMA creation or he just used an existing organisation (perhaps pushed it a bit in the proper direction) is unknown.
I've done more research on this and although it seems that no in-universe confirmation has been given this theory is plausible and quite popular.
In this blog post the author speculates that every person brought to the island was brought because of Jacob's actions:
Obviously, the clearest indication is the Man in Black's literal
accusation that Jacob brings people to the Island. The former is
referring specifically to the Black Rock when he says: "You brought
them here." But I take the Man in Black to mean that Jacob is always
the one who brings people to the Island, including DHARMA
and mentions some facts to support this claim.
Even more facts are listed in one of the theories on Lostpedia:
The DHARMA Initiative may have been brought to the Island by Jacob.
This would appear to be contradicting with the fact that Jacob is
normally associated with the Hostiles/Others. However, it could be
related to the ongoing rivalry between Jacob and his nemesis.
DHARMA maybe had more knowledge
on the Temple and Monster than the Hostiles. This explains why they
build the Barracks over the summoning chamber, why they were the only
ones who had a name for the monster, why the Temple had a DHARMA logo,
why the countdown timer had hieroglyphs, why Horace Goodspeed would
build a cabin for Jacob, why Paul would have an Ankh necklace, and why
the Frozen Wheel has the shape of an octagon, like the DHARMA logo.
An interesting opinion is given in this Reddit discussion:
Now, some of the DI were candidates to replace Jacob, so maybe you
could say he brought those specific people there, or maybe he just
chose candidates from those that he knew would end up on the island
anyway. Hard to say whether they were part of the plan (especially
when the future is all set in stone, and Jacob can apparently see into
In fact we don't know for sure that there was a candidate from DHARMA, but it's also not a prove, that there wasn't.
Lastly I would like to address the fact that as you wrote the DHARMA's role in the plot was quickly shoved under the carpet:
In my opinion the DHARMA background and their work (including Valanzetti equation) are not that important to the plot. I would see DHARMA more like an expanded plot device to introduce us to The Others, then through The Others to the secrets of The Island and finally to the Jacob/MiB story.