First good read man and thx.
Not every sharingan is a rinnegan.both got unique dojutsu different to other one.
Think kishi named it also as a rinnegan coz Itz the next step of the rinnegan. Like 3T , MS , EMS all called as a sharingan in common. After sharingan it's rinnegan.sasuke got advance version of the rinnegan so simply can't call it as a shringan.
I think sasuke went beyond the Normal rinnegan coz sage chakra+Indra chakra.
Sage not having the same eye is also fair coz it's not something kaguya born with and we also saw Indra and ashura didnt get the same power as sage. So when it passing down it's getting downgraded.
Then there come the problem that how that first IT was stopped?
May be sage brother got that 3rd eye from mother? Coz he must have something from mother not only sage. Or kaguya just died becoming victim of the fruit?
Now assume that kaguya had the biyakugan in birth will make her a hyuuga.
It's hard to write more on phone and hope you can understand above bro.