They were one of the two strongest clans.
At the time they were thinking of doing the Coup . . .
They were nothing more than security guards.
Two Senju were Hokage. . then they chose someone from a 'lesser house'
[neither Senju nor Uchiha . . . ]
Then second Hokage from a lesser house [The 4th was neither Senju nor Uchiha]
They obviously had no representation on the Council
[I'm sure an Uchiha on the council would have warned them]
I think Itachi was they 'great hope' to have a Uchiha as Hokage
but . .. they kind of did not think the other would let it happen