From the hints we got, it's safe to assume Zoro is a Shimotsuki. I mean that's not like the greatest family and it doesn't take away his strength or anything, we already know that he achieved most of his strength through training. With that said, being a Shimotsuki doesn't mean Zoro is destined and born as someone great who should be the greatest because we are learning about his family and bloodline that's it, nothing more nothing less.
If the Shimotsuki Bloodline, like people say will make Zoro as someone who is destined for something great then why is Yasuie not special?
We have got plenty of clues and hints like Zoro having the same Blood type as Ryouma, same Birth month and the birth month(which is November) literally Translates to Shimotsuki, "the month of frost". Zoro, Ryouma and Shimotsuki Ushimaru(Zoro's theorized father) are almost spitting images of each other.
Zoro means Fox in Spanish and Ushimaru had a pet fox(however that connects lol). Zoro is said to be a Japanese If looked at a real world perspective in an Sbs by Oda, and we all know that land of Wano Kuni is Japanese based.
Zoro rarely connects with People but he immediately had Yasuie and Toko following him and it's known that Yasuie is a shimotsuki while Toko is theorized to be one too. He also met Onimaru who was a pet of Shimotsuki, also Kawamatsu told about Shimotsuki Ushimaru to Zoro after exploring the land of Ringo where Shimotsuki clan situated.
Notice how most Shimotsuki related things in Wano is centered around Zoro? Forget about these because in some Sbs when Oda was talking about Shimotsuki Kozaburo he hinted Zoro's lineage immediately too. By far the most obvious thing would be Zoro being a Shimotsuki and maybe not knowing it himself or he just doesn't care about it maybe that's what "Sunachhi" was all about.