The sisterhood is an aspiration in some feminist circles, part of the problem in others, and just not part of the equation in most.
That's the thing with feminism - it's an umbrella term for dozens of ways of viewing the problems faced by girls and women and potential solutions, many of them at odds with each other. For most, I don't think sisterhood is as aspiration so saying we're 'meant to uplift each other' isn't going to ring true for many, whether or not they view themselves as feminists.
By “sisterhood” I mean something closer to women supporting each other, not viewing each other as rivals and not centering male approval in how we treat each other.
Those are three separate things that are being treated as inherently connected. They are sometimes, but it's not as inherent as the posts sounds.
Just because I'm not supporting another woman doesn't mean I view them as rivals or centreing male attention. Just because another woman doesn't support me, disrespects me, or is otherwise negative towards me doesn't mean she views me as a rival or doing it for male attention.
Just because someone treats me as a rival doesn't mean they don't support me in some areas or are treating me as a rival for male attention. I work in a female-dominated workplace, I have had colleagues who viewed me as competition against themselves, their friend, or their mentee - I have been in direct open competition with some of them at times - but they still supported me in other work. Life's complicated like that.
Just because I prioritise certain men over most women doesn't mean I would treat another woman badly, just like my prioritising certain women over most men doesn't mean I would treat another man badly. We can seek another person's approval or prioritise their happiness over another's without being an ass about it.