Ah, crud. When I was referring to 'Lilith', I was referring to Rei 3's full soul, which is seeking to reunite with her larger angelic body. He cries out to Rei, rejecting the fact that she made a different choice to him.
Yes. Gendo does have a strong relationship with Yui, but it really does begin to show off as selfish especially when Rei's motherly instincts towards Shinji prompt her to disallow Gendo from instigating instrumentality. Why? He abandoned Shinji. Now he only wishes for Shinji to finish what he started. But this is the turning point. We're finally introduced to his true intentions, and the nature of his actions, especially how Yui views them. It's basically confirmed in the manga that he's going to die without being able to reunite with Yui. In the anime, that's because he's literally chomped in half. In the manga, it's because he was shot by Ritsuko at the last second. He's going to die without Shinji helping him, like how Shinji allowed Misato to join him in instrumentality by turning her into LCL (even though she died already). Gendo is just left there.
Now this is the most important part: Yui says that Shinji symbolises the love between Gendo and Yui. Yui's basically saying that by rejecting Shinji, he's basically rejecting Rei, Yui and Lilith (because they are practically all one and the same, considering that Yui's motherly instincts have transferred to Rei). Although, in the end, right after this, he remembers the first time he touched Shinji's hand and the manga says that his 'soul was saved'.
He has strong devotion towards Yui, but it's laser-guided. He doesn't see anything but his wife, and rejected Shinji, who symbolised their love. He accepts punishment. He rejects the notion that Rei/Yui's motherly instincts have rejected him. He definitely did not want to take Shinji in his paradise, and he only ever truly contemplated it in the final moments of his life.
