Can't activate feature because of an existing column in a child site

Problem :

• Manage Content and Structure shortcut not show at Site Actions
• Need activate feature 
• SharePoint Server Publishing Infrastructure
• SharePoint Server Publishing
• Activate publishing feature but fail

Error message :

When activating the publishing feature :

Enable-SPFeature : The field with Id {51d39414-03dc-4bd0-b777-d3e20cb350f7} defined in feature {aebc918d-b20f-4a11-a1db-9ed84d79c87e} was found in the current site collection or in a subsite.

Resolution : From different source 
http://sharepoint.stackexchange.com/questions/31597/cant-activate-feature-because-of-an-existing-column-in-a-child-site
http://sharepoint.stackexchange.com/questions/27196/activating-ootb-features-fails

Summary of resolution :
Run the script to found out which sub site to delete :

It seems like your problem is that a site column with ID is already exist at root or sub sites. This site is created typically by SharePoint publishing infrastructure and it corresponds to PublishingStartDate site column. This is (typically) installed by a hidden FEATURE when publishing feature is activated.

Use following script to find the webs in which it exists:

 $url = Read-Host -Prompt "Provide Url for the web";
$fieldID = "51d39414-03dc-4bd0-b777-d3e20cb350f7"
$site = Get-SPSite $url
foreach ($web in $site.AllWebs)
{   
    foreach ($field in $web.Fields)
    {
        if ($field.Id -eq $fieldID)
        {
            $msg = [String]::Format("{0} contains the field.", $web.Url);
            Write-Host $msg
        }
    }
}
 2. Run the Poweshell script :


 $urlSite = "http://server/sitecollection"
$urlSubWeb = "http://server/sitecollection/subweb" #sub web identify by above step
$tempPath = "d:\temp"
$featId = "f6924d36-2fa8-4f0b-b16d-06b7250180fa" # Publishing feature
# Backup the site collection in case something goes wrong


Backup-SPSite -Identity $urlSite -Path "$tempPath\sitecoll.bak"

Export-SPWeb -Identity $urlSubWeb -Path "$tempPath\subweb.cmp" -IncludeUserSecurity -IncludeVersions All

Remove-SPWeb  -$url SubWeb

Enable-SPFeature -Identity $featId -Url $urlSite 
after enable the Feature still unable to resolve the issues: Run below command

Manually deactivate the both feature 

Run Below command :

stsadm -o activatefeature -name PublishingSite -url $url -force 

stsadm -o activatefeature -name PublishingWeb -url $url -force 

stsadm -o activatefeature -name PublishingResources -url $url -force

Lastly manually activate the feature  and test on it.

#final step restore the sub site
New-SPWeb -Url $urlSubWeb -Language 1033

Import-SPWeb -Identity $urlSubWeb -Path "$tempPath\subweb.cmp"  -IncludeUserSecurity