Could you explain the proximity arguments for the Estermonts and Florents, please? Why does marrying a family member into the royal family give you a claim on the throne? And if the Estermonts and Florents have a claim through proximity (presumably from their marriages to Steffon and Stannis), shouldn’t Cersei also have a (better) claim through her marriage to Robert?

So here’s my way of thinking, and there’s no reason necessarily to think it’s better than anyone else’s (thinking of @nobodysuspectsthebutterfly, @goodqueenaly, and @warsofasoiaf here):

If what matters is who the closest blood kin of the king is, then the Florents are kin to Princess Shireen and inlaws to Stannis (if you accept that Stannis and Shireen were the lawful kings/heirs to the Iron Throne). Next most recently, the Estermonts are kin to Robert, Stannis, and Renly through Cassana Estermont, regardless of how you see the succession going from that point on. After that, the Targaryens are kin to Robert, Stannis, Renly through Rhaelle Targaryen,  as well as to all previous kings. 

Cersei’s blood relationship to Robert Baratheon is much more distant than any of those, going back 90 years Gowen Baratheon and Tya Lannister, but even then that pairing died without successful issue, and 120 years ago when an unnamed Baratheon woman married a male Lannister and had issue. It’s highly unclear whether Cersei has any blood connection to Robert.

Now, there is an argument that she’s the mother of Joffrey and Tommen, but…A. their parentage and kingship was publicly put in question and Cersei has already admitted to incest in one case, and B. claims rarely go up the generations in that way, and C. even if the Lannister branch was chosen, Targaryen and Great Council precedent would suggest that a male cousin rather than Cersei herself, would inherit.