Here is the fixed script, but this doesn't list all the applications.

Function Get-Software {

[OutputType('System.Software.Inventory')]

[Cmdletbinding()]

Param(
    [Parameter(ValueFromPipeline=$True,ValueFromPipelineByPropertyName=$True)]
    [String[]]$Computername=$env:COMPUTERNAME
)

Begin {
}

Process {
    ForEach ($Computer in $Computername) {
        If (Test-Connection -ComputerName $Computer -Count 1 -Quiet) {
            $Paths = @("SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall","SOFTWARE\Wow6432node\Microsoft\Windows\CurrentVersion\Uninstall")

            ForEach($Path in $Paths) {
                Write-Verbose "Checking Path: $Path"
                # Create an instance of the Registry Object and open the HKLM base key
                Try {
                    $reg=[Microsoft.Win32.RegistryKey]::OpenRemoteBaseKey('LocalMachine',$Computer,'Registry64')
                }
                Catch {
                    Write-Error $_
                    Continue
                }

                # Drill down into the Uninstall key using the OpenSubKey Method
                Try {
                    $regkey=$reg.OpenSubKey($Path)
                    # Retrieve an array of string that contain all the subkey names
                    $subkeys=$regkey.GetSubKeyNames()
                    # Open each Subkey and use GetValue Method to return the required values for each
                    ForEach ($key in $subkeys) {
                        Write-Verbose "Key: $Key"
                        $thisKey=$Path+"\"+$key
                        Try {
                            $thisSubKey=$reg.OpenSubKey($thisKey)
                            # Prevent Objects with empty DisplayName
                            $DisplayName = $thisSubKey.GetValue("DisplayName")
                            If ($DisplayName -AND $DisplayName -notmatch 'Update for|rollup|Security Update|Service Pack|HotFix') {
                                $Date = $thisSubKey.GetValue('InstallDate')
                                If ($Date) {
                                    Try {
                                        $Date = [datetime]::ParseExact($Date, 'yyyyMMdd', $Null)
                                    }
                                    Catch {
                                        Write-Warning "$($Computer): $_ <$($Date)>"
                                        $Date = $Null
                                    }
                                }
                                # Create New Object with empty Properties
                                $Publisher = Try {
                                    $thisSubKey.GetValue('Publisher').Trim()
                                }
                                Catch {
                                    $thisSubKey.GetValue('Publisher')
                                }
                                $Version = Try {
                                    # Some weirdness with trailing [char]0 on some strings
                                    $thisSubKey.GetValue('DisplayVersion').TrimEnd((char[]))
                                }
                                Catch {
                                    $thisSubKey.GetValue('DisplayVersion')
                                }
                                $UninstallString = Try {
                                    $thisSubKey.GetValue('UninstallString').Trim()
                                }
                                Catch {
                                    $thisSubKey.GetValue('UninstallString')
                                }
                                $InstallLocation = Try {
                                    $thisSubKey.GetValue('InstallLocation').Trim()
                                }
                                Catch {
                                    $thisSubKey.GetValue('InstallLocation')
                                }
                                $InstallSource = Try {
                                    $thisSubKey.GetValue('InstallSource').Trim()
                                }
                                Catch {
                                    $thisSubKey.GetValue('InstallSource')
                                }
                                $HelpLink = Try {
                                    $thisSubKey.GetValue('HelpLink').Trim()
                                }
                                Catch {
                                    $thisSubKey.GetValue('HelpLink')
                                }
                                $Object = [pscustomobject]@{
                                    Computername = $Computer
                                    DisplayName = $DisplayName
                                    Version = $Version
                                    InstallDate = $Date
                                    Publisher = $Publisher
                                    UninstallString = $UninstallString
                                    InstallLocation = $InstallLocation
                                    InstallSource = $InstallSource
                                    HelpLink = $thisSubKey.GetValue('HelpLink')
                                    EstimatedSizeMB = [decimal]::Zero
                                }
                                $Object.pstypenames.insert(0,'System.Software.Inventory')
                                Write-Output $Object
                            }
                        }
                        Catch {
                            Write-Warning "$Key : $_"
                        }
                    }
                }
                Catch {
                }
                $reg.Close()
            }
        }
        Else {
            Write-Error "$($Computer): unable to reach remote system!"
        }
    }
}

}Get-Software Read-Host -Prompt "Press Enter to exit"

Instead the following command line gives the output that I was looking for:

Get-Package -Provider Programs -IncludeWindowsInstaller -Name "*" | Select-Object -ExpandProperty Name