First i must admit that i've never used ControlState to persist values across postbacks.
But control developers who create Controls that are used by other developers might want to use it to ensure functionality even if ViewState is disabled. Otherwise disabling would result in incorret behaviour or errors that are not obvious.
So in my opinion you would use ControlState if you would normally use ViewState but you must ensure that the control also works when ViewState is disabled.
In this case the sentence "Do not use control state as an alternative to view state" would be wrong, because you should use ViewState as well only for small amounts of data that must be persisted across postbacks.
An example to clarify what i mean:
Consider you want to create a custom control that inherits GridView. You can store the data in ViewState to maintain values across postbacks. You should allow to disable the ViewState on your control without any problems, you should not rely on the fact that it's enabled, because developers who are using your control might want to reduce network traffic. But you could provide other properties that are stored in ControlState like SortDirection, PageIndex, DeleteButtonText etc. that you think are important and are single values, therefor less resource intensive than storing all GridViewRows.
Edit: According to your actual problem:
You could enable ViewState for some controls and disable it for others, but you cannot disable it for whole page and enable it for child-controls in this page.
If you set turn page's ViewState off, then there is no way for you to
enable ViewState for specific components. This is because ViewState is
serialzed recursively, so when if the Page is not allowing ViewState,
it will not serialize the ViewState for any of it's child controls. If
you don't want to explicitly turn ViewState off on individual
controls, but want to keep some controls ViewState aware, the best way
would be writing a small utility method which turns ViewState off for
all controls (using recursion or otherwise). Then enable ViewState for
the controls that you would like to enable ViewState for.
Alternatively, a middle ground and less forceful way may possible if
controls are groups inside other container controls (such as Panel).
You can disable ViewState for all controls inside a Panel by disabling
ViewState of the Panel.
Enable ViewState for few controls and disable for others/page
Edit: Here is an extension to enable/disable ViewState for all child-controls(VB.NET)
<System.Runtime.CompilerServices.Extension()> _
Public Sub EnableChildViewState(ByVal parentControl As System.Web.UI.Control, enable as Boolean)
If parentControl.HasControls Then
For Each c As System.Web.UI.Control In parentControl.Controls
c.EnableViewState = enable
EnableChildViewState(c, enable)
Next
End If
End Sub
You can call it:
Page.EnableChildViewState(False)
MyControl.EnableViewState = True
