While influence could be argued of who had more military power which clan held the most monetary power for most of the period?
Ōuchi clan.
They were doing the exclusive trade with Korea.
They are called many educated nobles from Kyoto for working to enhance cultural.
And culture that their've created is called "Ouchi culture".Also there are still a lot of historic sites, including the national treasure to the Yamaguchi area now.
Your question is whether the wealth will lead to military force, but 31st generation family head Yoshitaka Ōuchi was ruined due to leave the military in excellent busho Harukata Sue.Then they were destroyed in 1557.