Hinata is from a noble clan that carries great blood, already with this a similarity can be drawn between Hinata and Naruto/Sasuke that would not exist with Sakura. She is born as being the Main Heir to the Hyuga Clan, but in time is neglected by her father due to her being too gentle and not wanting to fight. This would give her a very good goal to strive for, just as Naruto strived for Hokage and Sasuke for vengeance. It certainly would have held more depth than Sakura’s goal to be Sasuke’s Girl Friend.
As she is a Hyuga, she also posses the Byakugan. Both her Clan and her Blood Line Limit would make it much easier for Kishimoto to give her power ups, as she would be from the same family line as Naruto and Sasuke. Lion Chakra Avatars, Chakra Senbons, the Tenseigan [or something like so], White Flames, Chakra blades, Air Manipulation, Cutting Chakra Tubes, etc…. All of these would have been realistic, possible power up options for a Hyuga Main Heroine.
The Hyuga’s Byakugan, Fighting Style, and Chakra Expulsion Abilities would lay the groundworks for many of such techniques.
Her Clan and Blood Line Limit would also guarantee her a spot as being permanently relevant alongside the two other Main characters, unlike with Sakura, where Kishimoto struggled to keep her relevant for even 5 chapters in a row, with every one of her moments always being shoehorned in.
Her Role as Heir, the Byakugan’s secrets, The Curse Seal and the Caste System of the Hyuga all had potential to be used for well done character development for Hinata & her relationship with her Father, Sister, Cousin, and Clansmen. It also would have been rather good development for Naruto and Sasuke, as, by observing Hinata’s relationship with her clan, they would come to realize that even those with living families can have struggles.
WIth Naruto and Sauske being of Ootsutsuki blood, and Kakashi possessing powers from this lineage for 99% of the manga, Sakura sticks out like a massive sore thumb in this team. Due to this, it would have been many times more appropriate to have all three genins in Team 7 be of Ootsutsuki decent, as opposed to having 3 of them plus a random nobody.
Likewise, Sasuke is Indra’s Incarnation whilst Naruto is Ashura’s. Hinata is already the Incarnation of Hamura in the canon story line, again showcasing she would have had far more relevance to the Ootsutsuki Family plot line, which is pretty much all of the latter half of part 2.
Her backgorund would also make it easier for her to realistically understand how Naruto and/or Sasuke feels and for her to develop deep, meaningful bonds with them, as many aspects of her childhood is similar to both of them in different ways.
Naruto never had anyone to love him. Sasuke lost those he loved. Hinata was rejected by the ones she loved. Hinata has very similar qualities to these two, far more than Sakura ever did, as is shown during the many, many times in which she fails to understand either Naruto or Sasuke. Anyone who writes knows this. Her story just goes perfectly with Sasuke’s and Naruto’s story.
Naruto & Hinata were both considered failures, and they were more often than not rejected and insulted for who they were. Despite this, they both tried their best to prove others wrong and aimed to change themselves. Both also aim to hold a high status within Konoha, with Naruto aiming to be Hokage and Hinata aiming to be the Hyuga Clan’s Leader.
Hinata being from the Hyuga and Naruto being from the Uzumaki could have led to some good development, in that the Uzumaki are skilled in Fuinjutsu, so a scenario could have arisen in which Naruto and Hinata develop a Fuinjutsu together which functions similar to the Caged Bird Seal, except that it wouldn’t be capable of damaging the brain cells, plus it would be branded on all Hyugas as opposed to just the Branch Family. Such a scenario would have provided proper development for the Uzumaki Clan’s techniques, the Hyuga Clan’s politics, Hinata’s role as an important figure within her clan, and it would fulfill Naruto’s promise of changing the Hyuga Clan to Neji [which he failed to do in canon]
You can connect Sasuke and Hinata in so many ways. Both are from Clans that posses Dojutsus. Their pasts are also very similar in that both aimed to please their fathers, only to be compared to their superior sibling [Itachi & Hanbi]. Because of this, both aimed to get stronger, and in due time, both were able to surpass their siblings in skill and power. Sasuke & Hinata also had bad blood between a relative of theirs [Itachi & Neji], but eventually, were able to come to good terms with each other.
Likewise, Hinata being from the Hyuga and Sasuke being of the Uchiha would also have provided the opportunity for interesting moments and interaction between them, such as them training together to improve their dojutsu abilities, at least the ones they share in common. Not to mention Hinata posses both Fire and Lightening style, which also would open up the window for her to have relevant training sessions with Sasuke and/or Kakashi, and in turn, help to develop the bonds between the three.
It could have led to things such as the relationship between the Hyuga and Uchiha and how clan politics & clan relationships work within the Leaf Village. Perhaps it would have led to Sasuke viewing not only Naruto as a rival, but Hinata also, as she posses a Dojutsu & same Nature elements just as he does.
All of these similarities that Hinata shares with Naruto and/or Sasuke would have made her a much more fitting member as the third main character.