List Styles & Style Sets

Since you already have such code, this is more for others who may stumble across this question. Here is  revised code from my Add-In.

Sub OutlineStyleCopy1()
'
' StyleCopy Macro written by Charles Kyle Kenyon 14 November 2001, revised 12 March 2017, revised April 2026
'
' Copies all Built-In Heading Styles and the List Style using them to the current document
' If the numbering is attached to styles other than the Built-In Headings, the style names need to be set below
' Copyright 2001, 2017, 2026 All rights reserved
' Copies styles from stylesheet global template to active document
'   keyboard shortcut Ctrl+Shift+Alt+3
'   Thank you to Hans Vogel for the assistance with the workaround needed for a false alert about changes to the attached template
'       (often the normal template)
' This is set to copy a numbering list style "Headings" and associated paragraph styles using the Organizer Method
' This method could be used to create a StyleSheet to transfer non-numbered styles (like the BodyStyleCopy1 procedure)but
'   in Word 2007 and later the [Quick] Style Sets do this very nicely. A List Style, though, can not be part of such a set.
'
' Updated to declare and use variable names for the Styles being copied to make for easier changes.
'   As set, this uses the built-in heading styles associated with a List style.
'   The headings can have different local names, which are associated with a .NameLocal.
'   If using other paragraph level styles, the styles must exist in this Add-In template before running the macro.
'   If a List Style is involved, it must be set up using the procedures shown in Shauna Kelly's page. The variable
'   sListStyle must be changed if using a different List Style.
' If not transfering a List Style, comment out the last Organizer line dealing with a List Style.
'
    Dim sThisTemplate As String, sTargetDoc As String, sListStyle As String
    Dim sStyle1 As String, sStyle2 As String, sStyle3 As String, sStyle4 As String, sStyle5 As String ' Style Names
    Dim sStyle6 As String, sStyle7 As String, sStyle8 As String, sStyle9 As String ' Style Names
    Dim i As Integer    ' Counter 1 - use for copying styles three times
    Dim iCount As Integer    ' Counter 2 - use to loop copying of heading styles
    Dim rResponse As Variant    ' can be vbMsgBoxResult in Word 2000 or later
    Dim bSaved As Boolean ' Hans Vogel
    '
    Let bSaved = ActiveDocument.AttachedTemplate.Saved
    '
    Let sListStyle = "Headings" ' change if List Style is different name
    '
    ' ASSIGN Names for styles attached to List - must be exact!
    ' The ones below give the names for the Built-In Heading Styles
    Let sStyle1 = ThisDocument.Styles(wdStyleHeading1).NameLocal
    Let sStyle2 = ThisDocument.Styles(wdStyleHeading2).NameLocal
    Let sStyle3 = ThisDocument.Styles(wdStyleHeading3).NameLocal
    Let sStyle4 = ThisDocument.Styles(wdStyleHeading4).NameLocal
    Let sStyle5 = ThisDocument.Styles(wdStyleHeading5).NameLocal
    Let sStyle6 = ThisDocument.Styles(wdStyleHeading6).NameLocal
    Let sStyle7 = ThisDocument.Styles(wdStyleHeading7).NameLocal
    Let sStyle8 = ThisDocument.Styles(wdStyleHeading8).NameLocal
    Let sStyle9 = ThisDocument.Styles(wdStyleHeading9).NameLocal
' Error Handler set
    On Error GoTo NoDocument    ' Call when no document is open.
' Define This Template and Target Document variables
    sThisTemplate = ThisDocument.FullName    ' name and path of global template
    sTargetDoc = ActiveDocument.FullName    'generates error if no document open
' If any other errors, continue on
    On Error Resume Next
    rResponse = MsgBox(Prompt:="This command redefines your " _
                               & vbCrLf & "Heading Styles 1-9. Are you sure you want to do this?" _
                               & vbCrLf & vbCrLf & "If you are not sure, answer 'No' and make a backup of your document." _
                               & vbCrLf & "Then run the command to copy the styles again.", _
                       Title:="Are you sure you want to redefine your styles?", _
                       Buttons:=vbYesNo + vbExclamation)
    If rResponse = vbNo Then Exit Sub
    ' Copy Heading Styles to Active Document and Headings List Style
    For i = 1 To 3
        With Application
            '   Changed to use Word constants to make language independent
'            .OrganizerCopy Source:=sThisTemplate, _
'                           Destination:=sTargetDoc, Name:=ThisDocument.Styles("Headings").NameLocal, _
'                           Object:=wdOrganizerObjectStyles    ' Headings List Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle1, Object:=wdOrganizerObjectStyles    ' Level 1 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle2, Object:=wdOrganizerObjectStyles    ' Level 2 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle3, Object:=wdOrganizerObjectStyles    ' Level 3 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle4, Object:=wdOrganizerObjectStyles    ' Level 4 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle5, Object:=wdOrganizerObjectStyles    ' Level 5 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle6, Object:=wdOrganizerObjectStyles    ' Level 6 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle7, Object:=wdOrganizerObjectStyles    ' Level 7 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle8, Object:=wdOrganizerObjectStyles    ' Level 8 Style
            .OrganizerCopy Source:=sThisTemplate, _
                           Destination:=sTargetDoc, Name:=sStyle9, Object:=wdOrganizerObjectStyles    ' Level 9 Style
            .OrganizerCopy Source:=sThisTemplate, _
                            Destination:=sTargetDoc, Name:=sListStyle, Object:=wdOrganizerObjectStyles ' ListStyle
        End With
    Next i
        rResponse = MsgBox(Prompt:="Do you also want to copy the accompanying Body Text Styles?", _
                       Buttons:=vbYesNo, Title:="Body Text) Styles Too?")
    If rResponse = vbNo Then Exit Sub
    Call BodyStyleCopy1
    Exit Sub    ' Skip error handler routine
NoDocument:        ' Error Handler - Will be called if there is not an open document
    MsgBox Prompt:="Sorry, this command is only available when you have a document open.", _
           Title:="No document open!", Buttons:=vbExclamation
End Sub

It copies a List Style set in the template with the associated paragraph styles. While I believe that the [Quick] Style sets can reliably transfer numbering, they do not transfer the list style, while this macro does. This mimics  what happens if you use the Organizer to copy a list style. When you do that, Word asks before copying each of the associated paragraph styles. In the macro, you are asked once.

For your purposes, it has surplus at the end about the Body Text styles. The Add-In has a Body Text style available for a following style for each of the Heading Styles. The code for adding those is a separate procedure in the Add-In. You could delete that from the macro, just answer "No," or create the Body Text styles. If you create the Body Text Styles, you would need that macro as well.

This will be on my downloads page in a few days. I added substantial commenting and set it so the Style names are set as variables at the beginning of the macro. 

List styles usually don’t get saved with a style set, so they won’t move to a new document.

You may need to create or copy the list styles again in the new document manually.

See: [Quick] Style Sets can include multilevel numbering attached to styles.

List styles cannot be a part of a [Quick] Style Set.  You can use a set to transfer the numbering scheme, though, which can be edited. The numbering needs to be attached to discrete paragraph styles which have to be in the set.

• Automatic Numbering in Word - my 5-page pdf
• Quick Style Sets and Themes in Microsoft Word
• [Quick] Style Sets with MultiLevel Numbering attached to paragraph styles (There is a zip folder containing 100+ [Quick] Style Sets with numbering and a ReadMe document.)

It is also possible to set up a macro that uses the Organizer method, but your client would have to install the template as an Add-In and allow macros to run. A Global StyleSheet in Microsoft Word?

You may have some luck if you set up your list style so that each level is also associated with a unique paragraph style. Add the paragraph styles to your style set.

Generally, though, my experience is that bullets and numbering are not working consistently with style sets.

As Kidd_Ip suggested, use a document template instead.

Thanks.  I can import list styles using the Organizer; and I have written macros that import sets of styles (along with list styles) into whatever document I'm editing (and those work perfectly).  When deploying this kind of thing for clients, it would just be a lot easier if I could give them style sets to import; but I need those list styles to be included.  I've frankly never understood why list styles cannot be made to show up in the styles pane or the style gallery.  Frustrating...

I believe not but using templates (.dotx) or themes rather than relying solely on style sets.

https://support.microsoft.com/en-gb/office/customize-or-create-new-styles-d38d6e47-f6fc-48eb-a607-1eb120dec563

https://www.technipages.com/how-to-access-and-use-style-sets-in-microsoft-word/