I think that Anne does recognize that link between her and Max - I think it manifests in her paranoia about Max's intentions and her conversations with Jack about protecting themselves from her.
When her relationship with Max starts, she resists - partly, yes, from her own internalized homophobia and defensiveness, but also because she is painfully aware of Max's efforts to tie herself to Anne to protect herself. Anne can't stop herself from pursuing it anyway, because she can also see that she herself needs this, but when she speaks to Jack about keeping their heads clear when it comes to Max I think it's with the clear understanding that Max is in a position where she needs to wrest some power for herself and Anne worries it will happen by Max dividing the two of them (which, of course, it does).
And it's not necessarily textual, but I think that Anne sees herself in that. I think that's part of why she's so immediately defensive, immediately aggressive towards Max despite their romance, because she knows what this is and why it's happening. She knows, yes, that Max schemes and places herself in positions to be needed to secure her own safety from her experience with Max up until that point, but she also knows what it looks like to hold on desperately to a lifeline, and she knows that desperation is dangerous to her and Jack precisely because she felt it. I think it would also be reasonable to say that it brings up some guilt about her own role in inserting herself in Jack's life, worsening her insecurity about Jack staying around (which is of course partly why leaving helped her security in her and Jack's relationship so much - knowing that she can survive without him and he without her sheds some of that fear).
I think that's actually a main throughline in their relationship. Max loves Anne, is devoted to her, but fundamentally what she is (like Silver) is a survivalist - she protects her own interests and is always guarded on some level to everyone around her, including Anne. She seduces Anne when Anne needed to explore that side of her, she gives Anne purpose when Jack chooses the venture over her, she protects Anne's interests with the gold - but her own comfort? Her own safety? Max is the only one who can take care of Max. We can see that guard over and over again. Even when she is clearly passionately in love with Anne, she keeps that distance, insisting on parting ways while Anne pleads with her to take that wall down and stay.
And Anne knows this. But she's in love with Max. She can't break down that wall that Max puts up to protect herself, can't stop the momentum that Max builds towards survival, but knows she can't pull away and protect herself either. It's why she was so frustrated after Max betrayed her. She saw it coming, she knew it could happen, she knew she should hate Max with every fibre of her being - but she can't. Just like she couldn't stop herself from letting Max insert herself into her life in the first place.
I think that's why "there is nothing important that does not include you" is so impactful on Anne. Now that I'm thinking about it, I don't actually think it's the gesture, the refusal of the one thing Max has been striving for for the benefit of repairing things with Anne. It's a large part of it, for sure, but Max has done things for Anne before. She's taken care of Anne, she's maneuvered her survival strategies to bring Anne into them before. No, what makes this moment different is that it is the first moment of actual, true vulnerability Max expresses to Anne in the whole show. There is no manipulation, there is no wall, there is no next steps to ensure her survival. We all know it works out, but in that moment, Max has nothing, is asking for nothing, other than Anne.
I think that the whole show builds their relationship up to that - Anne gaining her independence and Max breaking down her walls, but also their ability to be with each other without relying on each other for some aspect of their survival: Anne with her need to be loved in the way Max can (as further evidenced by the fact that Anne can then also presumably find it with Mary Read), Max with her need to attach herself to something secure.