Next Gen Sand Siblings (NGSS from now on) are so much more compelling if you interpret them as adopted; either canon or DoS. There’s just something fascinating about the idea that the position of Kazekage is restricted to a clan and yet the oldest child of the previous Kazekage has married into a clan of a different village entirely. It’s all the fun of aristocratic inheritance/successions but with ninja!
As I touched on in Our Share of the World, it’s probably Shinki that has the most resentment for Shikadai and even his is kind of tempered by the knowledge that his biggest competition is his younger sister. Yodo, I think, probably just doesn’t like Shikadai as a person–his possible claim on the position of Kazekage has nothing to do with it. Araya and Shikadai are friendliest by default.
I feel like, considering the Council favors Yodo over Shinki (even though, as I mentioned, he’s the one with the blood limit) that they’re not too focused on the matter of blood lines at this point. Especially since Shikadai has been raised as a Leaf shinobi. Though probably, earlier on when he was just born or even a little before then, there were negotiations to decide on that exactly.
They probably conceded to Shikadai being raised Nara because they figured that either Gaara or Kankurou would have biological children eventually (hahahahahaha, nope) and then by the time they began worrying, Gaara began adopting the NGSS and then a whole other round of debates were instigated on whether or not they were valid candidates and, if so, which would be the better one.
I can see them possibly considering an arranged marriage another generation down the line or some kind of Kazekage clan repopulation thing like what’s going on with the Uchiha repopulation in Walking Around (Like Regular People). But I also feel like, given the trend towards more liberal stances of marriages and families and that Temari has already no doubt flouted the Council’s authority and married Shikamaru despite their likely protests, an arranged marriage won’t hold up unless the NGSS that becomes Kazekage is just like, yeah okay why not.
(You may want to check out this post thread speculating how various Kages may be chosen in the Elemental Nations, which started with me wondering about Shikadai’s possible claim to the Kazekage position. This was before I discovered the NGSS existed, of course, but still relevant.)