Unite multiple shapes they become one shape. All the individual fills are ignored and a single fill is applied to the united object.
That gradient you have can't be accomplished on a single shape as a fill, other than using a gradient mesh. But when using the
Unite command, meshes aren't generated for you.
A better option may be using a stroke with a gradient applied to the stroke.
You could then choose
Object > Expand Appearance on the stroke and a shape with a mesh will be generated.
Be aware though, a mesh shape is not as easily handled (altered, edited) as a standard shape in many cases. It's often easier to just leave it as a stroke.
(this was all done in CS6 as well)