I think part of the challenge for this cast is one of getting into a familiar groove.
The original TOS cast were consummate professionals with already years of experience in their craft. That with doing a series day after day for months on end helped them settle into their roles quickly and getting into their groove so to speak. The cast here have diversions and other lives besides full time acting. That might account for some awkwardness in their performances.
I agree that Uhura could have fulfilled the McKennah role in TOS. And considering the subject matter it would have been additionally poignant for Uhura (as a black women in the 1960s) to act as the go-between with Lolani.