I doubt the Sarutobi dynasty would be closer to making the 'the Hokage must be one of us' claim than the Senju were. Granted, it's not impossible that the Senju could've made that claim but chose not to, and thus the Sarutobi could also be in a position to make the claim despite not having a stronger claim than the Senju, but I still don't think it's likely:
The core part of Konoha ending up a dynastic monarchy is exclusivity against people outside of the dynasty coming into power, and between Hiruzen and Asuma we have Minato and Jiraiya. What I could see happening is one of Asuma's children becoming groomed for being Hokage because Sarutobi ninjas are clearly Hokage material, but I feel like it would take another generation or two of uninterrupted Sarutobi before people start believing that being Sarutobi is necessary to be Hokage.