It's a simple who takes the position first. Senju took the position first meaning Uchihas were destined to sit back seat to the Senju from that point on.
Shodai Hokage: Hashirama Senju, the beginning of Senju supremacy and Uchiha inferiority
Nidaime Hokage: Tobirama Senju, younger brother of Shodai.
Sandaime Hokage: Hiruzen Sarutobi, student of former two Hokage, no Uchiha influence there
Yondaime Hokage: Minato Namikaze, a little round about but still senju influence, the student of the Third Hokage's student so still within that same general influence
Godaime Hokage: Student of Third Hokage, granddaughter of first, grand neice of second, she has everybody except Minato.
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The mighty Sharingan was blinded, which wasn't surprising.
Their first mistake was turning their back on their leader, which made him rage like an emo and try to kill Hashirama, which promoted immedient untrust between the two groups AGAIN, however if you look in depth Madara's attack was just an excuse to be used by the Senju so they could make sure no uprising would come again. No Uchiha would've became Hokage if they didn't get First Hokage and the same would've gone for the Senju.
The tradition seems to be that, the student or close relations within the Hokage liniage is what determines the next Hokage. You can be OP like Itachi but it doesn't matter unless your ties are strong with one of the Hokage in some way (see examples above)
Jiraiya first offered position of Godaime, Tsunade eventually took the place although most people (on the base) see her as a weaker tier of the 5. Orochimaru was origionally up until Hiruzen said na, and then was looking at Minato, the pupil of his own pupil. People wanted Kakashi to take the position but thanks to plot no jutsu and Kakashi's humbleness (which I believe is true, he's not prime Kage material) he wasn't admitted as Hokage although he was about too. Danzo Shiruma, had ties with Tobirama in some way, and was besties with Hiruzen, although they had the love/hate kind of relationship but regardless you'd never see an Uchiha as a Hokage candidate.
They were killed off in Sandaime's second term, leaving only two left, one was out as a spy so cross him out of the mix, the other raged and left, although so there you go.
That's just how Kishi decided to leave it, Senju=superior, Uchiha=inferior once Konoha's founding.
THat is perhaps the only time Senju and Uchiha weren't deadlocked and there was a clear winner since the two sons of the Sage. Just like before, Senju was chosen over Uchiha.