Unfortunately BSplineFunction has many shortcomings. It does not honour exact input, it does not obey to PiecewiseExpand, it does not provide the parameterization of BSplineFunction, and now we discover that it does not play well with RevolutionPlot3D. Here is an alternative:

myBSplineFunction[pts_?MatrixQ] := 
  Module[{knots, deg}, 
   knots = Rationalize@BSplineFunction[pts][[6, 1]]; 
   deg = BSplineFunction[pts][[3, 1]];
   Apply[Function, 
    List@Sum[
      pts[[i + 1]] BSplineBasis[{deg, knots}, i, #], {i, 0, 
       Length[pts] - 1}]]];
RevolutionPlot3D[
 Evaluate[myBSplineFunction[controlPoints][t]], {t, 0, 1}]

With ParametricPlot we don't need to separate the components:

ParametricPlot[BSplineFunction[controlPoints][t], {t, 0, 1}]

Here it is RevolutionPlot3D to be blamed more for the inconvenience.

Thank you. This is what I was looking for as it wasn't immediately obvious to me that the components needed to be separated out. Appreciate the help!