I think a big factor would be prejudice against AI in general. I forget the name of the TNG episode where Data was assigned as the captain of an individual ship in a task force dealing with a Romulan conflict. Data's first officer was reluctant to serve under an Android Captain -- Data essentially had to prove his competence by winning the engagement.
Extend that in general to have an entire ship run by AI. That first officer can't be alone in his sentiments (in fact we *know* he's not: Commander Maddox in "Measure of a Man" tried to have Data disassembled entirely!) Based on that, I'm sure there would be a lot of resistance to having computer control of any major command function.