Computer Name not Showing

That's because of the last line and the Format-Table used there. The Format-* cmdlets are meant to format screen output, not further processing. They should usually be the last element of the pipeline.
The most convenient is usually an export as csv, so that you can easily process it further or view it in Excel or whatever:

$array | Where-Object { $_.DisplayName } | select ComputerName, DisplayName, DisplayVersion, Publisher | Export-Csv -NoTypeInformation "C:\DGS Tools\Scripts\logs\installedapps.txt"

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If you insist on the table view, you'll have to send the output through Out-String before passing it onto the the file:

$array | Where-Object { $_.DisplayName } | select ComputerName, DisplayName, DisplayVersion, Publisher | ft -auto | Out-String -Width 1024 | Out-File "C:\DGS Tools\Scripts\logs\installedapps.txt"\installedapps.txt"

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I like the CSV output, but it still doesn't show the computer name. I attached the script I ran this time as InstalledAppsNew.txt and the output I received
installedapps.csv
InstalledAppsNew.txt

That is because you changed line 6 from " $computername = $pc" to "$computername = $pc.Computername".
Change that back, or remove the line completely and change the ForEach in line 4 to use $computer instead of $pc:

foreach ($computer in $computers) {

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Without the $pc.computername I get:

Exception calling "OpenRemoteBaseKey" with "2" argument(s): "The network path was not found.
"
At C:\DGS Tools\Scripts\InstalledApps.ps1:14 char:5
+     $reg=[microsoft.win32.registrykey]::OpenRemoteBaseKey(‘LocalMachine’,$comput ...
+     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    + CategoryInfo          : NotSpecified: (:) [], MethodInvocationException
    + FullyQualifiedErrorId : IOException
 
You cannot call a method on a null-valued expression.
At C:\DGS Tools\Scripts\InstalledApps.ps1:18 char:5
+     $regkey=$reg.OpenSubKey($UninstallKey)
+     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    + CategoryInfo          : InvalidOperation: (:) [], RuntimeException
    + FullyQualifiedErrorId : InvokeMethodOnNull
 
You cannot call a method on a null-valued expression.
At C:\DGS Tools\Scripts\InstalledApps.ps1:22 char:5
+     $subkeys=$regkey.GetSubKeyNames()
+     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    + CategoryInfo          : InvalidOperation: (:) [], RuntimeException
    + FullyQualifiedErrorId : InvokeMethodOnNull

No matter how I run it. I attached what I've run with changing out to $computer. It's the same output as above
InstalledAppsNew.txt

Question answered.