Ah, all right. I think that I did talk about this briefly in one analysis, but I’ll go more into depth here. I’m writing this on mobile, so apologies for the lack of pictures and formatting and double apologies if autocorrect fudges anything up!
In Neon Genesis Evangelion, SEELE believes that Instrumentality and the return to Lilith is the correct path for humanity to take. In NGE2, the second-tier canon video game that was heavily influenced and partially written by the Project EVA staff, we witness SEELE’s version of Instrumentality, wherein they utilise Tabris/Nagisa Kaworu/Adam (as opposed to Ayanami Rei/Lilith) as the quantum messenger. Under SEELE’s command, the souls are harvested, with the souls of the members of SEELE surviving as immortal beings while the rest of humanity dies. In the anime, Ikari Yui opposes their goals and intends for an Instrumentality for everyone, which suggests that SEELE did it not plan to include all of humanity, which in turn supports NGE2’s explanation of SEELE’s intentions to be canonical. However, SEELE’s dialogue in End of Evangelion appears to suggest otherwise; the film portrays SEELE as a deeply religious organisation who sincerely believed in Instrumentality, perhaps because the members’ own hearts were empty and desiring death/escapism. I view SEELE as a combination of both: Given Yui’s commentary, I think that either SEELE initially intended the selves-as-gods Instrumentality and later changed their minds (after Yui’s death), OR that SEELE intended the selves-as-gods Instrumentality but recognised that plans had shifted as of End of Evangelion, prompting SEELE to simply be content with being tanged themselves.
The director’s cut of the twenty-fourth episode includes a very misleading conversation between Kaworu and SEELE. During the conversation, SEELE claims to support the angels as the true successors and assures Kaworu that the members do not plan to utilise Unit 01 as their private ark. Yet SEELE is lying blatantly to Kaworu in an attempt to manipulate Kaworu into taking out Gendo (re: NGE2). Kaworu, of course, listens very patiently to them, nods, smiles, and proceeds to dive into Terminal Dogma because the Fifth Children is suicidal and intends to return to being one with Adam because of that heart, overflowing with sorrow, as Rei puts so eloquently in End of Evnagelion. Anyway, the reason that I bring this up is because SEELE lies directly to Kaworu about the support of the angels and Gendo’s purpose. Hence, it is reasonable to interpret that SEELE likewise lied about not wanting to use Unit 01 as a private ark to ride out Instrumentality. Ikari Yui actually ends up using Unit 01 as an ark and becomes immortal in End of Evangelion, but I’ve already commented on Yui’s grey morality.
Now, Gendo. Gendo intended to use Rei to control Instrumentality; that’s why he implanted Lilith’s soul into the body and raised the First Children to be perfectly obedient to him. That’s also why he split Lilith’s soul in two, because a partial soul is less responsive and volatile than a full soul, hence, obedience. His specific goal was to reunite with Yui and spend eternity with her. This implies that he only wanted partial Instrumentality, so that he could still conceptualise the difference between himself and Yui. In any case, Rei III more or less gave him the metaphorical finger, took in Adam/Kaworu’s soul, and returned to Lilith.
In the Rebuilds/new theatrical versions, SEELE is implied in the third film to no longer have corporeal forms and to have somehow watched over humanity for quite a while. While the Illuminati nature of SEELE may explain the latter, the former is a bit more questionable. SEELE does not seem to be antagonistic and appears to wish for Instrumentality again, this time genuinely believing that Instrumentality and/or infinity is the way to go. It’s unclear if SEELE would prefer the former or the latter, but based upon the anime, it’s most probably the former.
Regarding the latter: The third film revealed that the Third Impact caused the proliferation of EVA-based life, being the combination of the Fruits of Life and Knowledge possessed by the angels and lilim, respectively. Because the Third Impact was stopped, the remains of the failed Impact are known as the failures of infinity.
As for Gendo’s plan, all we know is that he intends to reunite with Yui, that he lied to SEELE and purposefully interfered with both SEELE and Kaworu/Shinji’s plans in order to take out SEELE and Kaworu, and that he will put his true plan in motion in the fourth film.
(Kaworu/Shinji, by the way, intended to use the Lances of Longinus and Cassius to in some manner reverse or undo the effects of Third Impact. The most likely explanation is that Kaworu wanted to use Shinjiwielding the Fruit of Knowledgeto cause another Impact, this time lilim-based, since Impacts also terraform the planet. The effects of the Lance of Cassius are unknown at this time. Basically, Kaworu decided to give SEELE and NERV metaphorical middle fingers and fly away with Shinji in the sunset. Except that Kaworu was grossly misinformed, both on the Lances and on Shinji’s status as a F.A.R.)
We won’t actually be able to answer these questions until the fourth film comes out. So ask again then, and I’m sorry for being inconclusive!